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Questions and Answers
Which of the following equations can be solved by extracting the square root?
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.
The equation x² = -16 can be solved by extracting the square root.
False (B)
What values of x satisfy the equation x² = 36?
What values of x satisfy the equation x² = 36?
6, -6
For the equation y² = 81, the solution for y is _____.
For the equation y² = 81, the solution for y is _____.
Match the following quadratic equations with their solutions:
Match the following quadratic equations with their solutions:
Which method is appropriate for solving the quadratic equation when it is in the form $x^2 = c$?
Which method is appropriate for solving the quadratic equation when it is in the form $x^2 = c$?
The quadratic formula can be used for all forms of quadratic equations.
The quadratic formula can be used for all forms of quadratic equations.
Name one method used to solve quadratic equations.
Name one method used to solve quadratic equations.
To solve a quadratic equation by completing the ________, we manipulate the equation to form a perfect square.
To solve a quadratic equation by completing the ________, we manipulate the equation to form a perfect square.
Match each method of solving quadratic equations with its description:
Match each method of solving quadratic equations with its description:
Which of the following statements best describes a polynomial?
Which of the following statements best describes a polynomial?
An expression containing a variable in the denominator can still be classified as a polynomial.
An expression containing a variable in the denominator can still be classified as a polynomial.
What is a variable in algebra?
What is a variable in algebra?
In the expression $3x^2 + 4y$, the number 3 is referred to as the ______.
In the expression $3x^2 + 4y$, the number 3 is referred to as the ______.
Match the following terms with their definitions:
Match the following terms with their definitions:
Which of the following equations is a quadratic equation?
Which of the following equations is a quadratic equation?
The standard form of a quadratic equation is ax² + bx + c = 0.
The standard form of a quadratic equation is ax² + bx + c = 0.
Write the equation 3x² - 2x = -7 in standard form.
Write the equation 3x² - 2x = -7 in standard form.
A __________ equation is characterized by the highest degree being 2.
A __________ equation is characterized by the highest degree being 2.
Match the following equations to whether they are quadratic or not:
Match the following equations to whether they are quadratic or not:
Which type of polynomial has three terms?
Which type of polynomial has three terms?
A quadratic equation has the highest exponent of 2.
A quadratic equation has the highest exponent of 2.
What is the standard form of a linear equation?
What is the standard form of a linear equation?
The polynomial with two terms is called a ______.
The polynomial with two terms is called a ______.
Match the following equations with their types:
Match the following equations with their types:
What is the result when c < 0 in the equation $x^2 = c$?
What is the result when c < 0 in the equation $x^2 = c$?
If c = 0, then the equation $x^2 = c$ has one real root.
If c = 0, then the equation $x^2 = c$ has one real root.
What is the first step in solving a quadratic equation by factoring?
What is the first step in solving a quadratic equation by factoring?
To apply the zero-product property, each factor of a factored quadratic equation must be set equal to _____.
To apply the zero-product property, each factor of a factored quadratic equation must be set equal to _____.
Match the following outcomes with the corresponding conditions for $x^2 = c$:
Match the following outcomes with the corresponding conditions for $x^2 = c$:
What is the GCF of the expression $6x^2 + 15x^4$?
What is the GCF of the expression $6x^2 + 15x^4$?
The expression $y^2 + 10y + 25$ can be factored as $(y+5)(y+5)$.
The expression $y^2 + 10y + 25$ can be factored as $(y+5)(y+5)$.
Factor the expression $4x^2 - 12x + 9$.
Factor the expression $4x^2 - 12x + 9$.
The general form $x^2 - bx + c$ when factored gives the form of ______.
The general form $x^2 - bx + c$ when factored gives the form of ______.
Match the following expressions with their factors:
Match the following expressions with their factors:
Which of the following is a correct factorization of the trinomial $3x^2 + 14x + 16$?
Which of the following is a correct factorization of the trinomial $3x^2 + 14x + 16$?
The solutions to the equation $(y - 2)(y - rac{1}{2}) = 0$ are $y = 2$ and $y = - rac{1}{2}$.
The solutions to the equation $(y - 2)(y - rac{1}{2}) = 0$ are $y = 2$ and $y = - rac{1}{2}$.
What are the roots of the equation $x^2 + 5x + 6 = 0$?
What are the roots of the equation $x^2 + 5x + 6 = 0$?
The equation $(x + 4)(x + 4) = 0$ has a double root at _____.
The equation $(x + 4)(x + 4) = 0$ has a double root at _____.
Match the following equations with their solutions:
Match the following equations with their solutions:
Which of the following is the correct factorization of the expression $x^2 - 64$?
Which of the following is the correct factorization of the expression $x^2 - 64$?
The expression $25m^2 - 100$ can be factored as $(5m + 10)(5m - 10)$.
The expression $25m^2 - 100$ can be factored as $(5m + 10)(5m - 10)$.
What is the first step in factoring a trinomial of the form $x^2 + bx + c$?
What is the first step in factoring a trinomial of the form $x^2 + bx + c$?
The difference of two squares can be factored as () and ().
The difference of two squares can be factored as () and ().
Match the following expressions with their corresponding factorizations:
Match the following expressions with their corresponding factorizations:
Which of the following represents the standard form of a quadratic equation?
Which of the following represents the standard form of a quadratic equation?
The equation $x^2 - 4x + 4 = 0$ can be factored as a perfect square trinomial.
The equation $x^2 - 4x + 4 = 0$ can be factored as a perfect square trinomial.
What values of $a$, $b$, and $c$ are in the quadratic equation $2x^2 + 5x - 3 = 0$?
What values of $a$, $b$, and $c$ are in the quadratic equation $2x^2 + 5x - 3 = 0$?
The equation $3x^2 - 12 = 0$ can be solved by __________ method.
The equation $3x^2 - 12 = 0$ can be solved by __________ method.
Match the following equations with their factored forms:
Match the following equations with their factored forms:
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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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