In the quadratic equation a x^2 + b x + c = 0, the coefficients a, b, and c are real numbe… — ACT

In the quadratic equation $a x^2 + b x + c = 0$ , the coefficients $a$ , $b$ , and $c$ are real numbers, and $a = \frac{2}{3}$ . If one of the roots of this equation is $x = \frac{1}{2} - i\sqrt{2}$ , where $i = \sqrt{-1}$ , what is the value of the product $bc$ ?

$-1$ Answer
B
$-\frac{7}{9}$
C
$-\frac{1}{2}$
D
$0$
E
$\frac{7}{9}$

Answer

-1

For any quadratic equation with real coefficients, if a complex number is a root, its conjugate must also be a root. Thus, the two roots are

x_1 = \frac{1}{2} - i\sqrt{2}

and

x_2 = \frac{1}{2} + i\sqrt{2}

. Using Vieta's formulas, the sum of the roots is

x_1 + x_2 = 1 = -\frac{b}{a}

, which gives

b = -\frac{2}{3}

. The product of the roots is

x_1 x_2 = \frac{1}{4} - 2i^2 = \frac{9}{4} = \frac{c}{a}

, which gives

c = \frac{3}{2}

. The product

bc

is therefore

(-\frac{2}{3})(\frac{3}{2}) = -1

Step-by-Step Solution

Find the second root of the quadratic equation.

The second root is

x_2 = \frac{1}{2} + i\sqrt{2}

Since the quadratic equation has real coefficients, complex roots must occur in conjugate pairs.

Calculate the sum of the roots to find the coefficient

b

b = -\frac{2}{3}

The sum of the roots is

x_1 + x_2 = (\frac{1}{2} - i\sqrt{2}) + (\frac{1}{2} + i\sqrt{2}) = 1

. By Vieta's formulas,

x_1 + x_2 = -\frac{b}{a}

. Substituting

a = \frac{2}{3}

gives

1 = -\frac{b}{2/3} \implies b = -\frac{2}{3}

Calculate the product of the roots to find the coefficient

c

c = \frac{3}{2}

The product of the roots is

x_1 x_2 = (\frac{1}{2} - i\sqrt{2})(\frac{1}{2} + i\sqrt{2}) = (\frac{1}{2})^2 - (i\sqrt{2})^2 = \frac{1}{4} - 2i^2

. Since

i^2 = -1

, the product is \frac{1}{4} - 2(-1) = \frac{9}{4}

. By Vieta's formulas,

x_1 x_2 = \frac{c}{a}

. Substituting

a = \frac{2}{3}

gives \frac{9}{4} = \frac{c}{2/3} \implies c = \frac{3}{2}

Calculate the product

bc

bc = -1

Multiplying the values of

b

and

c

gives

bc = (-\frac{2}{3})(\frac{3}{2}) = -1

Key Concept

Relating the roots of a quadratic equation with real coefficients to its coefficients via Vieta's formulas and the Complex Conjugate Root Theorem.

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