Answer each question. Find the values of a, b, and c for which the quadratic equation. ax2+bx+c=$$0ax^2 + bx + c = 0 $$has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.)−3,2-3, 2

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 96

Chapter 2, Problem 96

Answer each question. Find the values of a, b, and c for which the quadratic equation. $a x^{2} + b x + c = 0$ has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.)
$negative 3 comma 2$

Verified step by step guidance

Recall that if a quadratic equation has solutions (roots) $ r_1 $ and $ r_2 $, then it can be written in factored form as $ a(x - r_1)(x - r_2) = 0 $.

Given the solutions $ -3 $ and $ 2 $, substitute these values into the factored form: $ a(x - (-3))(x - 2) = 0 $, which simplifies to $ a(x + 3)(x - 2) = 0 $.

Expand the factors $ (x + 3)(x - 2) $ by using the distributive property (FOIL method): $ x \cdot x + x \cdot (-2) + 3 \cdot x + 3 \cdot (-2) $.

Simplify the expression from the previous step to get a quadratic expression in standard form: $ x^2 + (\text{sum of coefficients})x + (\text{constant term}) $.

Identify the coefficients $ a $, $ b $, and $ c $ from the expanded quadratic expression $ a x^2 + b x + c = 0 $. Note that $ a $ can be any nonzero constant, often taken as 1 for simplicity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Equation Standard Form

A quadratic equation is expressed as ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. Understanding this form is essential because it defines the structure of the equation whose roots or solutions we seek.

Zero-Factor Property

The zero-factor property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is used to find solutions of quadratic equations by factoring and setting each factor equal to zero.

Forming Quadratic Equations from Roots

Given the roots of a quadratic equation, you can reconstruct the equation by writing it as (x - root1)(x - root2) = 0 and then expanding. This reverse process helps find the coefficients a, b, and c based on the given solutions.

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Solving Quadratic Equations by the Square Root Property

Related Practice

Textbook Question

Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | 4x² - 23x - 6 | = 0

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Textbook Question

Answer each question. Find the values of a, b, and c for which the quadratic equation. $a x^{2} + b x + c = 0$ has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.)

$4 comma 5$

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Textbook Question

Solve each inequality. Give the solution set using interval notation. 3x+6 / x-5 > 0

Solve each inequality. Give the solution set using interval notation. (x+7) / (2x+1) ≤1

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Textbook Question

Solve each inequality. Give the solution set using interval notation. $\frac{3 x - 2}{x} - 4 > 0$

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Answer each question. Find the values of a, b, and c for which the quadratic equation. ax2+bx+c=0ax^2 + bx + c = 0 has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.)−3,2-3, 2

Verified video answer for a similar problem:

Key Concepts

Quadratic Equation Standard Form

Zero-Factor Property

Forming Quadratic Equations from Roots

Answer each question. Find the values of a, b, and c for which the quadratic equation. $a x^{2} + b x + c = 0$ has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.)
$negative 3 comma 2$