Lesson 2: Solving Quadratic Equations
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Questions and Answers

Which of the following equations can be solved by extracting the square root?

  • x² - 4x + 4 = 0
  • x² = -5
  • x + 3 = 0
  • x² = 25 (correct)

    • The equation x² = -16 can be solved by extracting the square root.

    False

    What values of x satisfy the equation x² = 36?

    6, -6

    For the equation y² = 81, the solution for y is _____.

    <p>±9</p> Signup and view all the answers

    Match the following quadratic equations with their solutions:

    <p>x² = 16 = x = ±4 y² = 25 = y = ±5 m² = 49 = m = ±7 n² = 100 = n = ±10</p> Signup and view all the answers

    Which method is appropriate for solving the quadratic equation when it is in the form $x^2 = c$?

    <p>Extracting square roots</p> Signup and view all the answers

    The quadratic formula can be used for all forms of quadratic equations.

    <p>True</p> Signup and view all the answers

    Name one method used to solve quadratic equations.

    <p>Extracting square roots, Factoring, Completing the squares, or Quadratic formula</p> Signup and view all the answers

    To solve a quadratic equation by completing the ________, we manipulate the equation to form a perfect square.

    <p>squares</p> Signup and view all the answers

    Match each method of solving quadratic equations with its description:

    <p>Extracting square roots = Used when $x^2 = c$ Factoring = Rewriting the equation as a product of binomials Completing the squares = Transforming the equation into a perfect square trinomial Quadratic formula = Using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$</p> Signup and view all the answers

    Which of the following statements best describes a polynomial?

    <p>An algebraic expression that represents a sum of terms.</p> Signup and view all the answers

    An expression containing a variable in the denominator can still be classified as a polynomial.

    <p>False</p> Signup and view all the answers

    What is a variable in algebra?

    <p>A letter used to represent a number.</p> Signup and view all the answers

    In the expression $3x^2 + 4y$, the number 3 is referred to as the ______.

    <p>numerical coefficient</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Variable = A number without a variable Constant = Letters representing numbers in math Numerator = Top part of a fraction Denominator = Bottom part of a fraction</p> Signup and view all the answers

    Which of the following equations is a quadratic equation?

    <p>-7y² + 4y = 0</p> Signup and view all the answers

    The standard form of a quadratic equation is ax² + bx + c = 0.

    <p>True</p> Signup and view all the answers

    Write the equation 3x² - 2x = -7 in standard form.

    <p>3x² - 2x + 7 = 0</p> Signup and view all the answers

    A __________ equation is characterized by the highest degree being 2.

    <p>quadratic</p> Signup and view all the answers

    Match the following equations to whether they are quadratic or not:

    <p>7² + 2x² + 3 = 0 = Not Quadratic 2y³ - 3y⁴ - 2 = 0 = Not Quadratic 3m³ + 2m² = 0 = Quadratic -7y² + 4y = 0 = Quadratic</p> Signup and view all the answers

    Which type of polynomial has three terms?

    <p>Trinomial</p> Signup and view all the answers

    A quadratic equation has the highest exponent of 2.

    <p>True</p> Signup and view all the answers

    What is the standard form of a linear equation?

    <p>Ax + By = C</p> Signup and view all the answers

    The polynomial with two terms is called a ______.

    <p>Binomial</p> Signup and view all the answers

    Match the following equations with their types:

    <p>2x² + 3x + 4 = 0 = Quadratic equation 4y² - 3y + 4 = 0 = Cubic equation 2x + 3 = Expression x² + y² = 0 = Quartic equation</p> Signup and view all the answers

    What is the result when c < 0 in the equation $x^2 = c$?

    <p>There are no real roots</p> Signup and view all the answers

    If c = 0, then the equation $x^2 = c$ has one real root.

    <p>False</p> Signup and view all the answers

    What is the first step in solving a quadratic equation by factoring?

    <p>Transform the quadratic equation into standard form if necessary.</p> Signup and view all the answers

    To apply the zero-product property, each factor of a factored quadratic equation must be set equal to _____.

    <p>zero</p> Signup and view all the answers

    Match the following outcomes with the corresponding conditions for $x^2 = c$:

    <p>c = 0 = Two real roots c &gt; 0 = One real root c &lt; 0 = No real roots</p> Signup and view all the answers

    What is the GCF of the expression $6x^2 + 15x^4$?

    <p>3x^2</p> Signup and view all the answers

    The expression $y^2 + 10y + 25$ can be factored as $(y+5)(y+5)$.

    <p>True</p> Signup and view all the answers

    Factor the expression $4x^2 - 12x + 9$.

    <p>(2x-3)(2x-3)</p> Signup and view all the answers

    The general form $x^2 - bx + c$ when factored gives the form of ______.

    <p>(x-r)(x-s)</p> Signup and view all the answers

    Match the following expressions with their factors:

    <p>3m^2 - 27m - 90 = (m-30)(m+3) 69(z^2-3) = 69(z-3) 6z^2 - 6z + 9 = (6z-3)(6z-3) y^2 - 7y + 10 = (y-5)(y-2)</p> Signup and view all the answers

    Which of the following is a correct factorization of the trinomial $3x^2 + 14x + 16$?

    <p>(x + 6)(x + 2)</p> Signup and view all the answers

    The solutions to the equation $(y - 2)(y - rac{1}{2}) = 0$ are $y = 2$ and $y = - rac{1}{2}$.

    <p>False</p> Signup and view all the answers

    What are the roots of the equation $x^2 + 5x + 6 = 0$?

    <p>x = -2 or x = -3</p> Signup and view all the answers

    The equation $(x + 4)(x + 4) = 0$ has a double root at _____.

    <p>-4</p> Signup and view all the answers

    Match the following equations with their solutions:

    <p>$2x^2 + 2x + 6 = 0$ = No real solutions $x^2 + 5x + 6 = 0$ = x = -2, x = -3 $(y - 2)(y - rac{1}{2}) = 0$ = y = 2, y = 0.5 $(x + 3)(x + 1) = 0$ = x = -3, x = -1</p> Signup and view all the answers

    Which of the following is the correct factorization of the expression $x^2 - 64$?

    <p>(x + 8)(x - 8)</p> Signup and view all the answers

    The expression $25m^2 - 100$ can be factored as $(5m + 10)(5m - 10)$.

    <p>False</p> Signup and view all the answers

    What is the first step in factoring a trinomial of the form $x^2 + bx + c$?

    <p>Find two numbers that multiply to c and add to b.</p> Signup and view all the answers

    The difference of two squares can be factored as () and ().

    <p>sum, difference</p> Signup and view all the answers

    Match the following expressions with their corresponding factorizations:

    <p>x² - 49 = (x + 7)(x - 7) 16y² - 1 = (4y + 1)(4y - 1) x² - 25 = (x + 5)(x - 5) 9a² - 36 = (3a + 6)(3a - 6)</p> Signup and view all the answers

    Which of the following represents the standard form of a quadratic equation?

    <p>$x^2 + 3x + 2 = 0$</p> Signup and view all the answers

    The equation $x^2 - 4x + 4 = 0$ can be factored as a perfect square trinomial.

    <p>True</p> Signup and view all the answers

    What values of $a$, $b$, and $c$ are in the quadratic equation $2x^2 + 5x - 3 = 0$?

    <p>a = 2, b = 5, c = -3</p> Signup and view all the answers

    The equation $3x^2 - 12 = 0$ can be solved by __________ method.

    <p>extracting the square root</p> Signup and view all the answers

    Match the following equations with their factored forms:

    <p>$x^2 - 9 = 0$ = (x - 3)(x + 3) $y^2 + 6y + 9 = 0$ = (y + 3)^2 $x^2 - 4 = 0$ = (x - 2)(x + 2) $x^2 + 10x + 24 = 0$ = (x + 4)(x + 6)</p> Signup and view all the answers

    Study Notes

    Solving Quadratic Equations by Extracting Square Roots

    • Applicable when the equation is in the form (x^2 = C).
    • Example: From (x^2 - 16 = 0), derive (x = \pm 4).
    • Another example: From (y^2 = 81), derive (y = \pm 9).

    Steps for Solving Quadratic Equations

    • Begin with (b^2 + a^2 = 0) leading to ( \sqrt{b^2} = \pm \sqrt{a^2}).
    • For (m^2 = 1), deduce (m = \pm \sqrt{m}) yielding two potential values.
    • For a more complex equation like ((x - 4)^2 - 24 = 0):
      • Rearrange to ((x - 4)^2 = 24)
      • Find roots as (x = 9) and (x = -1).

    Quadratic Equation Overview

    • Four methods to solve:
      • Extracting square roots
      • Factoring
      • Completing the square
      • Quadratic formula
    • Standard quadratic form: (ax^2 + bx + c = 0).

    Identification of Quadratic Equations

    • Quadratic examples:
      • (7x^2 + 2x + 3 = 0) is NOT quadratic.
      • (3m^3 + 2m^2 = 0) is quadratic.
      • (-7y^2 + 4y = 0) is quadratic.
    • Not all equations with exponents are quadratic; degree matters.

    Writing Equations in Standard Form

    • Example transformations for standard form:
      • From (2x - 3x^2 + 1 = 0) to (3x^2 + 2x + 1 = 0).
      • Adjusting from (-7 + 4y^2 - 3y = 0) to (4y^2 - 3y - 7 = 0).

    Polynomials and Their Characteristics

    • Polynomials are algebraic expressions consisting of terms with non-negative integer exponents.
    • Essential requirements:
      • No negative exponents or variables in denominators.
    • Types of polynomials classified by the number of terms:
      • Monomial: 1 term
      • Binomial: 2 terms
      • Trinomial: 3 terms

    Factoring Techniques

    • GCF (Greatest Common Factor) method for factoring polynomials.
    • Example: (6x^2 + 15x^4) factors to (3x^2(2 + 5x)).
    • Perfect square trinomial: expressed as ((x + a)^2).

    Difference of Squares

    • The difference of squares is factored as the product of a sum and a difference:
      • Example: (x^2 - 100 = (x + 10)(x - 10)).

    General Trinomials

    • Recognizing factors of trinomials, both perfect squares and general forms:
      • Perfect square example: (y^2 + 10y + 25 = (y+5)^2).
      • General trinomial form example: (3m^2 - 27m - 90 = (m - 30)(m + 3)).

    Overview of Quadratic Equations

    • Quadratic equations are defined by the highest exponent being 2.
    • Reviewing methods for solving quadratic equations, including extraction, factoring, and writing in standard form.
    • Familiarize with specific examples and their transformations into standard form.

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    Description

    This quiz focuses on solving quadratic equations specifically by extracting the square root. It includes methods applicable only when the equation is in the form x² = C, providing examples and step-by-step processes for better understanding. Test your knowledge of this mathematical technique and enhance your problem-solving skills.

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