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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1. 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.
2. Given the solutions \( i \) and \( -i \), write the factored form of the quadratic as \( a(x - i)(x + i) = 0 \).
3. Use the difference of squares formula to expand \( (x - i)(x + i) \): \( (x - i)(x + i) = x^2 - (i)^2 \).
4. Since \( i^2 = -1 \), substitute this into the expression to get \( x^2 - (-1) = x^2 + 1 \).
5. 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 \( ax^2 + bx + c = 0 \).

Key Concepts

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.