For a constant k, the quadratic equation \frac{1}{3}x^2 - \frac{k}{6}x + (k - 10) = 0 has … — ACT

For a constant $k$ , the quadratic equation $\frac{1}{3}x^2 - \frac{k}{6}x + (k - 10) = 0$ has two real roots, $r_1$ and $r_2$ . If the sum of the reciprocals of the roots, $\frac{1}{r_1} + \frac{1}{r_2}$ , is equal to $-\frac{1}{2}$ , what is the value of $k$ ?

Answer: 7.5

Answer

The value of

k

7.5

By writing the sum of the reciprocals in terms of the sum and product of the roots, we find that

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}

. Substituting the values from Vieta's formulas (

r_1 + r_2 = \frac{k}{2}

and

r_1 r_2 = 3(k - 10)

) yields

\frac{k}{6(k - 10)} = -\frac{1}{2}

. Solving this equation for

k

results in

k = 7.5

Step-by-Step Solution

Identify the quadratic coefficients in terms of the constant

k

a = \frac{1}{3}

b = -\frac{k}{6}

, and

c = k - 10

Applying Vieta's formulas requires the coefficients from the standard form

ax^2 + bx + c = 0

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

r_1 + r_2 = \frac{k}{2}

and

r_1 r_2 = 3(k - 10)

Vieta's formulas state that the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

Express the sum of the reciprocals of the roots in terms of

k

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2} = \frac{k}{6(k - 10)}

Finding a common denominator allows the sum of the reciprocals to be written as the ratio of the sum of the roots to the product of the roots.

Equate the expression to

-\frac{1}{2}

and solve the resulting equation for

k

k = 7.5

Solving

\frac{k}{6(k - 10)} = -\frac{1}{2}

gives

2k = -6(k - 10) \implies 8k = 60 \implies k = 7.5

Key Concept

Vieta's Formulas and Algebraic Relationships of Roots

Estimated Time:2m 30s

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