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

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 98

Chapter 2, Problem 98

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

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Recall that if a quadratic equation has solutions (roots) $ r_1 $ and $ r_2 $, then it can be written as $ a(x - r_1)(x - r_2) = 0 $, where $ a $ is a nonzero constant.

Given the solutions $ i $ and $ -i $, write the factored form of the quadratic as $ a(x - i)(x + i) = 0 $.

Use the difference of squares formula to expand $ (x - i)(x + i) $: $ (x - i)(x + i) = x^2 - (i)^2 $.

Since $ i^2 = -1 $, substitute this into the expression to get $ x^2 - (-1) = x^2 + 1 $.

Therefore, the quadratic equation can be written as $ a(x^2 + 1) = 0 $. From this, identify $ a $, $ b $, and $ c $ by comparing to the standard form $ ax^2 + bx + c = 0 $.

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Key Concepts

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

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions (roots) of the equation are the values of x that satisfy it. Understanding the structure of quadratic equations is essential for finding coefficients given roots.

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. In reverse, if the solutions of a quadratic are known, the equation can be written as a product of factors set to zero, such as (x - r1)(x - r2) = 0, where r1 and r2 are roots.

Complex Conjugate Roots

When a quadratic equation with real coefficients has complex roots, they occur in conjugate pairs like i and -i. This means the quadratic can be expressed as (x - i)(x + i) = 0, which expands to x^2 + 1 = 0. Recognizing this helps in determining the coefficients a, b, and c.

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Complex Conjugates

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