In the quadratic equation 8x^2 - kx + 27 = 0, where k is a positive constant, one of the r… — SAT

In the quadratic equation $8x^2 - kx + 27 = 0$ , where $k$ is a positive constant, one of the roots is the square of the other root. What is the value of $k$ ?

Answer: 30

Answer

For the quadratic equation

8x^2 - kx + 27 = 0

, Vieta's formulas state that the product of the roots is

\frac{27}{8}

and the sum of the roots is

\frac{k}{8}

. Letting the roots be

r

and

r^2

, we obtain the equation

r^3 = \frac{27}{8}

, which gives

r = \frac{3}{2}

. The other root is

r^2 = \frac{9}{4}

. Summing these roots gives

\frac{3}{2} + \frac{9}{4} = \frac{15}{4}

. Equating this sum to the formula for the sum of the roots yields

\frac{k}{8} = \frac{15}{4}

, which simplifies to

k = 30

Step-by-Step Solution

Determine the product of the roots using Vieta's formulas.

The product of the roots is

r_1 r_2 = \frac{27}{8}

For any quadratic equation

ax^2 + bx + c = 0

, the product of its roots is given by

\frac{c}{a}

Use the root relationship to solve for one of the roots.

r_2^3 = \frac{27}{8} \implies r_2 = \frac{3}{2}

Substituting

r_1 = r_2^2

into the product equation yields

r_2^3 = \frac{27}{8}

Determine the value of the second root.

r_1 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}

The first root is the square of the second root.

Calculate the sum of the two roots.

r_1 + r_2 = \frac{9}{4} + \frac{3}{2} = \frac{15}{4}

The sum of the roots is needed to relate back to the linear coefficient of the quadratic equation.

Solve for the constant

k

using the sum of the roots.

\frac{k}{8} = \frac{15}{4} \implies k = 30

By Vieta's formulas, the sum of the roots is

-\frac{b}{a} = \frac{k}{8}

Key Concept

Quadratic Equations and Vieta's Formulas

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