Quadratic Equations Matching Quiz

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Questions and Answers

Match the following quadratic equations with their corresponding type of solution:

x² - 8x + 16 = 25 = Two real solutions 25x² - 8x = 12x - 4 = One real solution -x² + 4x = 13 = Imaginary solutions x² + 8x + 41 = 0 = No real solutions

Match the given equations with their solving methods:

x² + 3x = 5 = Quadratic Formula 4x² + 32x = -68 = Completing the square 6x(x + 2) = -42 = Factoring x² - 12x + 18 = Vertex form conversion

Match the following expressions with their perfect square trinomial form:

x² + 10x + c = c = 25, (x + 5)² w² + 13w + c = c = 42.25, (w + 6.5)² s² - 26s + c = c = 169, (s - 13)² x² + 30x + c = c = 225, (x + 15)²

Match the following quadratic equations with their corresponding forms:

<p>y = x² - 12x + 18 = Vertex form: y = (x - 6)² - 18 y = x² - 8x + 18 = Vertex form: y = (x - 4)² + 2 y = x² + 6x + 4 = Vertex form: y = (x + 3)² - 5 y = x² - 2x - 6 = Vertex form: y = (x - 1)² - 7</p> Signup and view all the answers

Match the following discriminant expressions with their result types:

<p>b² - 4ac = 0 = One real solution b² - 4ac &gt; 0 = Two real solutions b² - 4ac &lt; 0 = Imaginary solutions b² - 4ac = 1 = Two real solutions</p> Signup and view all the answers

Match the following quadratic expressions with their binomial factors:

<p>x² - 8x + 16 = (x - 4)(x - 4) x² + 4x + 4 = (x + 2)(x + 2) x² - 6x + 9 = (x - 3)(x - 3) x² - 10x + 25 = (x - 5)(x - 5)</p> Signup and view all the answers

Match the following broader concepts with their related equations or theorems:

<p>Completing the square = x² + 6x + 3 = 0 Quadratic Formula = x = (-b ± √(b² - 4ac)) / 2a Perfect Square Trinomial = x² + 30x + 225 Modeling with Quadratics = y = -16t² + 96t + 3</p> Signup and view all the answers

Match the given quadratic functions with their maximum height or time to hit the ground:

<p>y = -16t² + 96t + 3 = Maximum height = 150 feet y = x² - 12x + 18 = Time to hit ground = 6 seconds y = -16t² + 32t + 4 = Maximum height = 64 feet y = 2t² - 8t + 10 = Time to hit ground = 4 seconds</p> Signup and view all the answers

Match the following expressions of square roots of negative numbers with their corresponding values:

<p>√-25 = 5i √-72 = 6√2i √-9 = 3i √-54 = 6√6i</p> Signup and view all the answers

Match the following equations with their solutions for (x, y):

<p>2x - 7i = 10 + yi = (5, -7) x + 3i = 9 - yi = (6, -3) 9 + 4yi = -2x + 3i = (-5, 9/4) 2x + 3 = 7i = (2, 7)</p> Signup and view all the answers

Match the following components with their impedance in ohms:

<p>Resistor = R Inductor = Li Capacitor = -C Total Impedance in a Circuit = 5 - i</p> Signup and view all the answers

Match the following operations with their results in standard form:

<p>(8 - i) + (5 + 4i) = 13 + 3i (7 - 6i) - (3 - 6i) = 4 - 0i 13 - (2 + 7i) + 5i = 11 - 2i (6i) + (2 - 3i) = 2 + 3i</p> Signup and view all the answers

Match the following properties of square roots of negative numbers with their definitions:

<p>√-r = i√r (i√r)² = -r √-1 = i √-25 = 5i</p> Signup and view all the answers

Match the following components to their resistance or reactance values:

<p>Resistor = 5 ohms Inductor = 30 ohms Capacitor = 4 ohms Impedance = 5 - i</p> Signup and view all the answers

Match the following complex number operations with their correct answers:

<p>(8 - i) + (5 + 4i) = 13 + 3i (7 - 6i) - (3 - 6i) = 4 (3 + 2i) * (1 - 3i) = 9 - i (4 + i)(2 - i) = 8 + 2i</p> Signup and view all the answers

Match the following statements about square roots with their corresponding examples:

<p>Finding √-25 = 5i Finding √-72 = 6√2i Finding √-9 = 3i Finding √-54 = 6√6i</p> Signup and view all the answers

Match the discriminant conditions with the corresponding number and type of solutions:

<p>$b² - 4ac &gt; 0$ = Two real solutions $b² - 4ac = 0$ = One real solution $b² - 4ac &lt; 0$ = Two imaginary solutions $b² - 4ac ext{ undefined}$ = No solutions</p> Signup and view all the answers

Match the quadratic equations with their discriminant results:

<p>$a.x² - 6x + 10 = 0$ = $b² - 4ac &gt; 0$ $b.x² - 6x + 9 = 0$ = $b² - 4ac = 0$ $c.x² - 6x + 8 = 0$ = $b² - 4ac &lt; 0$ $4x² + 8x + 4 = 0$ = $b² - 4ac = 0$</p> Signup and view all the answers

Match the methods for solving quadratic equations with the appropriate scenarios:

<p>Graphing = Use when approximate solutions are adequate Using square roots = Use when solving $u² = d$ Factoring = Use when a quadratic can be easily factored Quadratic Formula = Can be applied to any quadratic equation</p> Signup and view all the answers

Match the quadratic equations with the type of solutions they result in:

<p>$ax² - 4x + c = 0$ = One real solution if $b² - 4ac = 0$ $ax² + 3x + c = 0$ = Two real solutions if $b² - 4ac &gt; 0$ $4x² + 8x + 4 = 0$ = One real solution $-5x² + 16 = -10x$ = Complex solutions if the discriminant is negative</p> Signup and view all the answers

Match the functions defining the height of an object with their scenarios:

<p>$h = -16t² + h₀$ = Object is dropped $h = -16t² + v₀t + h₀$ = Object is launched or thrown $h = -16t² + vt + h₀$ = Object is thrown at an angle $h = vt + h₀$ = Object is dropped from rest</p> Signup and view all the answers

Match the quadratic expressions with their respective quadratic formulas to use:

<p>$ax² + bx + c = 0$ = Quadratic Formula is applicable $u² = d$ = Use square roots Easily Factored Expression = Use factoring method $x² + 2x + 1 = 0$ = Use completing the square</p> Signup and view all the answers

Match the terms with their definitions in quadratic functions:

<p>$hâ‚€$ = Initial height (in feet) $vâ‚€$ = Initial vertical velocity (in feet per second) $t$ = Time in motion (in seconds) $h$ = Height of the object (in feet)</p> Signup and view all the answers

Match the pairs of values with the resulting nature of their quadratic solutions:

<p>$a = 1$, $c = 0$ = One real solution $a = 2$, $c = 8$ = Two imaginary solutions $a = 1$, $c = -1$ = Two real solutions $a = 4$, $c = 4$ = One real solution</p> Signup and view all the answers

Match the following equations with their respective solutions or interpretations:

<p>h = -16t² + 50 = Takes approximately 1.8 seconds to hit the ground 0 = -16t² + 50 = Find the zeros of the function h(1) - h(1.5) = Container fell 20 feet between 1 and 1.5 seconds R(2) = Maximum annual revenue of $968,000</p> Signup and view all the answers

Match the following algebraic expressions with their solving methods:

<p>x² + 3x + 2 = 0 = Solve by graphing (p - 4)² = 49 = Solve using square roots n² - 6n = 0 = Solve by factoring u² = -9u = Solve using any method</p> Signup and view all the answers

Match the following functions with their zeros:

<p>f(x) = x² - 8x + 16 = Zero at x = 4 g(x) = x² + 11x = Zero at x = 0 and x = -11 h = -16t² + 50 = Zero at t = √(50/16) x² - 9 = 0 = Zeros at x = 3 and x = -3</p> Signup and view all the answers

Match the following mathematical terms with their definitions:

<p>Complex Numbers = Numbers that include real and imaginary parts Imaginary Unit i = Defined as i = √-1 Rational Numbers = Numbers that can be expressed as the ratio of two integers Irrational Numbers = Numbers that cannot be expressed as a simple fraction</p> Signup and view all the answers

Match the following pairs of numbers with their categories:

<p>Natural Numbers = 1, 2, 3, 4, ... Whole Numbers = 0, 1, 2, 3, ... Integers = -1, 0, 1, 2, ... Real Numbers = All rational and irrational numbers</p> Signup and view all the answers

Match the complex number operations to their results:

<p>(9 - 2i)(-4 + 7i) = 49 + 26i (-3i)(10i) = -30 (8 - i)² = 63 - 16i (3 + i)(5 - i) = 20 + 2i</p> Signup and view all the answers

Match the following geometric problems with the necessary calculations:

<p>Area of rectangle = 36 = Find x in the expression x + 5 5 feet by 4 feet quilt = Remaining fabric for border Rectangular quilt border width = Add 10 square feet of fabric x + 2x = Total width of the quilt with border</p> Signup and view all the answers

Match the quadratic equations to their solutions:

<p>x² + 4 = 0 = x = ±2i 2x² - 11 = -47 = x = ±4 x² + 9 = 0 = x = ±3i x² + 49 = 0 = x = ±7i</p> Signup and view all the answers

Match the following functions with their maximum or minimum results:

<p>R(2) = -2000(2 - 24)(2 + 20) = Max annual revenue of $968,000 x² + 6x + 9 = Perfect square having minimum at x = -3 x² - 11x = -30 = Quadratic equation having two solutions h = -16t² + 50 = Height function with maximum initial height</p> Signup and view all the answers

Match the functions to their zeros:

<p>f(x) = 4x² + 20 = No real zeros f(x) = x² + 7 = No real zeros f(x) = -x² - 4 = No real zeros m(x) = x² - 27 = x = ±√27</p> Signup and view all the answers

Match the following steps in solving equations to their methods:

<p>Step involving √-1 = Definition of imaginary unit Using quadratic formula = Finding zeros of a quadratic function Set equation to zero = Extract potential solutions Factoring polynomials = Simplifying expressions to identify roots</p> Signup and view all the answers

Match the expressions to their forms:

<p>√-36 = 6i √-49 = 7i √-24 = 2√6i √-32 = 4√2i</p> Signup and view all the answers

Match the perfect square trinomials with their values of c:

<p>x² + 14x + c = c = 49 x² + 8x + c = c = 16 x² - 2x + c = c = 1 x² - 9x + c = c = 20.25</p> Signup and view all the answers

Match the operations with their resultant forms:

<p>(9 + 5i) + (11 + 2i) = 20 + 7i 7 - (3 + 4i) + 6i = 4 + 2i (16 - 9i) - (2 - 9i) = 14 - 0i (3 + i) + (5 - i) = 8 + 0i</p> Signup and view all the answers

Match each quadratic equation to the method used for solving:

<p>x² - 16x + 64 = 100 = Square Roots 3x² + 12x + 15 = 0 = Completing the Square x² - 10x + 7 = 0 = Completing the Square x² + 4x + 4 = 36 = Square Roots</p> Signup and view all the answers

Match the function equations to their characteristics:

<p>f(x) = 3x² + 6 = Parabola opening upward with vertex h(x) = 2x² + 72 = Parabola opening upward with vertex m(x) = x² - 27 = Parabola opening upward with zeros k(x) = 4x² + 20 = No real zeros</p> Signup and view all the answers

Match the following types of nonlinear systems with their corresponding methods of solving:

<p>System by graphing = Graphing the equations to find intersections System by substitution = Isolating one variable and substituting System by elimination = Adding or subtracting equations to eliminate a variable Quadratic inequalities = Finding regions defined by inequality conditions</p> Signup and view all the answers

Match the following quadratic inequalities with their correct forms:

<p>y ≥ 2x² = y is greater than or equal to a quadratic function y &lt; -x² + 1 = y is less than a quadratic function y ≤ x² + 7x + 6 = y is less than or equal to a quadratic function y &gt; 2x² - 3x - 6 = y is greater than a quadratic function</p> Signup and view all the answers

Match the following quadratic equations with the methods used to solve them:

<p>x² + y² = 16 = Graphing to find intersections with a line y = x² - 2x - 1 = Using substitution or graphing 2x² + 4x - y = -2 = Using elimination to isolate variable x + y = 4 = Substituting y into quadratic equation</p> Signup and view all the answers

Match the following quadratic inequalities with their graphical representation:

<p>y &lt; x² + 7x + 6 = Region below the parabola y ≥ 2x² = Region above or on the parabola y ≤ -x² + 1 = Region below or on the inverted parabola y &gt; 2x² - 3x - 6 = Region above the parabola</p> Signup and view all the answers

Match the following systems of equations with their solving method:

<p>y = -2x - 1 and y = x² - 2x - 1 = Solving by graphing x² + x - y = -1 and x + y = 4 = Solving by substitution 2r² - 3x - y = -5 and -x + y = 5 = Solving by elimination -3x² + 2x - 5 = y and -x + 2 = -y = Solving by elimination</p> Signup and view all the answers

Match the following systems with their outcomes:

<p>x² + y² = 4 and y = x + 4 = No solution; contradictory equations x² + y² = 1 and y = x + 1 = One solution at the intersection x² + y² = 16 and y = -x + 4 = Two solutions; intersection points x² + y² = 10 and y = -3x + 10 = One solution at the intersection</p> Signup and view all the answers

Match the following methods with their description:

<p>Graphing = Visual representation of equations Substitution = Replacing one variable with another Elimination = Removing one variable through addition or subtraction Inequalities = Finding ranges rather than specific points</p> Signup and view all the answers

Flashcards

Rational Number

A number that can be expressed as a fraction of two integers, where the denominator is not zero.

Irrational Number

A number that cannot be expressed as a fraction of two integers.

Imaginary Unit (i)

The square root of -1, denoted by the symbol 'i'.

Complex Number

A number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.

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Quadratic Equation

A quadratic equation is an equation that can be written in the standard form: ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' is not equal to 0.

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Zeros of a Function

The x-values where the graph of a function intersects the x-axis.

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Completing the Square

A technique for solving quadratic equations by rewriting the equation as a perfect square trinomial.

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Factoring

A technique for solving quadratic equations by factoring the expression into two linear factors.

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Square Root of a Negative Number

The square root of a negative number. For any positive real number r, √-r = i√r.

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Adding Complex Numbers

The process of combining two complex numbers by adding their corresponding real and imaginary parts.

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Subtracting Complex Numbers

The process of combining two complex numbers by subtracting their corresponding real and imaginary parts.

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Impedance in a Circuit

A number that represents the opposition to the flow of current in an electrical circuit. Usually measured in ohms (Ω).

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Multiplying Complex Numbers

The process of combining two complex numbers by multiplying them, following the distributive property and the rule i² = -1.

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Real Number

A number that only has a real part (no imaginary part).

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Imaginary Number

A number that only has an imaginary part (no real part).

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What is a complex number?

A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1).

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What is the standard form of a complex number?

The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the imaginary part.

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How do you multiply complex numbers?

To multiply complex numbers, distribute each term of the first number to each term of the second number, remembering that i² = -1.

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How to solve quadratic equations using square roots?

To solve a quadratic equation using square roots, isolate the squared term, then take the square root of both sides. Remember to consider both positive and negative solutions.

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What is a perfect square trinomial?

A perfect square trinomial is a trinomial that can be factored as (ax + b)², resulting in a square of a binomial.

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What is completing the square?

To complete the square, manipulate a quadratic expression by adding a constant term to both sides, making it a perfect square trinomial, thus allowing you to solve for the unknown variable by taking the square root.

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What are the zeros of a quadratic function?

The zeros of a quadratic function are the x-values where the graph intersects the x-axis. In other words, they are the values of x for which f(x) = 0.

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How do you find the zeros of a quadratic function?

To find the zeros of a quadratic function, set the function equal to zero and solve for x. You can use factoring, completing the square, or the quadratic formula.

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Discriminant

The expression b² - 4ac in the quadratic formula ax² + bx + c = 0, which determines the nature of the solutions.

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Quadratic Formula

The formula x = (-b ± √(b² - 4ac)) / 2a used to find the solutions (roots) of any quadratic equation ax² + bx + c = 0.

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Vertex of a Parabola

The highest point on the graph of a quadratic function, represented as (h, k) in vertex form.

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Vertex Form of a Quadratic Function

A way to rewrite a quadratic function y = ax² + bx + c in the form y = a(x - h)² + k, where (h, k) is the vertex.

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Solutions (Roots) of a Quadratic Equation

The values of x that make the quadratic equation ax² + bx + c = 0 true.

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Perfect Square Trinomial

The value of c needed to make an expression ax² + bx + c form a perfect square trinomial, which can be factored as (ax + b/2)².

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Analyzing the Discriminant

The value of the discriminant determines the number and type of solutions a quadratic equation has. If the discriminant is positive, there are two real solutions. If it's zero, there's one real solution. If it's negative, there are two imaginary solutions.

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Writing a Quadratic Equation

The process of finding the values of a and c in a quadratic equation to ensure a specific type of solution (e.g., one real solution, two real solutions).

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Height Function

Used to model the height of an object in motion considering its initial vertical velocity and height.

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Height Function (Dropped Object)

This function models the height of an object that is dropped from a certain height, ignoring initial velocity.

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Height Function (Launched Object)

This function models the height of an object that is launched or thrown, considering its initial velocity and height.

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Nonlinear system of equations

A system of equations where at least one equation is nonlinear (not a straight line).

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Solving by Substitution

A method for solving systems of equations where you isolate one variable in one equation and substitute it into the other equation.

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Solving by Elimination

A method for solving systems of equations where you manipulate the equations to eliminate one variable by adding or subtracting the equations.

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Quadratic Inequality in Two Variables

An inequality that can be written in the form y < ax² + bx + c, y > ax² + bx + c, y ≤ ax² + bx + c, or y ≥ ax² + bx + c, where a, b, c are real numbers and a ≠ 0. The graph represents a region of points.

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Solving a Quadratic Inequality Algebraically

A method to solve an inequality by isolating the variable and using the properties of inequalities. Involves considering possible cases and finding solution sets for each case that satisfy the original inequality.

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Solution to a System of Quadratic Inequalities

A point (x, y) that satisfies all inequalities in the system. These points lie within the overlapping shaded regions of the individual inequalities.

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Changing Inequality Symbols in a System

The inequality symbol(s) in a system of quadratic inequalities can be changed to create a new system that has different solutions. This changes the shaded regions which might include or exclude specific points.

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Study Notes

Algebra 2 Study Notes

  • Quadratic functions: Equations that can be written in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
  • Quadratic equation in one variable: An equation that can be expressed in standard form as ax² + bx + c = 0.
  • Real numbers and variables: Quadratic equations use real numbers and variables (often x) as coefficients and terms.
  • Roots of an equation: Solutions to a quadratic equation represent the x-values where the graph of the related function intersects the x-axis. This can be graphically or using algebraic methods.
  • Solving by graphing: Finding the x-intercepts of the related function (y = ax² + bx + c) to determine the roots. A graph is used to help visualize. A graphing calculator can help.
  • Solving by factoring: Writing the quadratic equation in factored form (ax² + bx + c = (px + q) (rx + s)) and utilizing the zero-product property to solve.
  • Solving by square roots: Manipulating the equation into the form u² = d, where u is an algebraic expression. Then take the square root of each side.
  • Zero-Product Property: If the product of two algebraic expressions equals zero, then at least one of the expressions must equal zero.
  • Finding Zeros of a quadratic function: The x-intercepts of the graph of a function f(x), also known as zero.
  • Equality of two complex numbers: Two complex numbers a + bi and c + di are equal if and only if a = c and b = d.
  • Complex Numbers: Combines real and imaginary numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = −1). Subsets of complex numbers include real numbers, imaginary numbers and pure imaginary numbers.
  • Adding and subtracting complex numbers: Add (or subtract) the real parts and imaginary parts separately.
  • Multiplying complex numbers: Using distributive or FOIL methods.
  • Solving quadratic equations by completing the square: To manipulate a quadratic equation and isolate x to solve.
  • Quadratic Formula: A formula for finding the roots (solutions) of any quadratic equation, ax² + bx + c = 0. The solution is x = (-b ± √(b² - 4ac)) / 2a
  • Example problems and practice exercises: These are present to illustrate practical application of the concepts covered to aid in the development of conceptual understanding.
  • Real-life problems: Applications involving quadratic functions, including projectile motion (e.g., the height of a dropped object). The maximum or minimum value of a quadratic function can be used to model a real-world problem.

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