In the quadratic equation 2x^2 - 11x + c = 0, where c is a constant, the ratio of the two … — ACT

In the quadratic equation $2x^2 - 11x + c = 0$ , where $c$ is a constant, the ratio of the two real solutions is $3:8$ . What is the value of $c$ ?

Answer: 12

The value of the constant

c

12

The correct answer is

12

. By representing the roots in the ratio of

3:8

3r

and

8r

, Vieta's formula for the sum of roots (

-\frac{b}{a}

) gives

3r + 8r = -\frac{-11}{2} \implies 11r = 5.5 \implies r = 0.5

. The actual roots are therefore

1.5

and

4

. Using Vieta's formula for the product of roots (

\frac{c}{a}

) gives

(1.5)(4) = \frac{c}{2} \implies 6 = \frac{c}{2} \implies c = 12

Represent the roots using the given ratio.

Let the two roots of the quadratic equation be

3r

and

8r

The ratio of the two solutions is specified as

3:8

Apply Vieta's formula for the sum of roots to find the ratio multiplier

r

3r + 8r = -\frac{-11}{2} \implies 11r = 5.5 \implies r = 0.5

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is given by

-\frac{b}{a}

Determine the numerical values of the two roots.

The roots are

3(0.5) = 1.5

and

8(0.5) = 4

Substitute the value of

r = 0.5

back into the expressions for the roots.

Apply Vieta's formula for the product of roots to solve for the constant

c

(1.5)(4) = \frac{c}{2} \implies 6 = \frac{c}{2} \implies c = 12

For a quadratic equation

ax^2 + bx + c = 0

, the product of the roots is given by

\frac{c}{a}

Vieta's formulas and the relationship between the roots and coefficients of a quadratic equation

Alternatively, you can express the roots using the quadratic formula:

x = \frac{11 \pm \sqrt{121 - 8c}}{4}

. Since the ratio of the smaller root to the larger root is

3:8

, we set up the equation:

\frac{11 - \sqrt{121 - 8c}}{11 + \sqrt{121 - 8c}} = \frac{3}{8}

. Cross-multiplying gives

8(11 - \sqrt{121 - 8c}) = 3(11 + \sqrt{121 - 8c}) \implies 88 - 8\sqrt{121 - 8c} = 33 + 3\sqrt{121 - 8c} \implies 55 = 11\sqrt{121 - 8c} \implies 5 = \sqrt{121 - 8c} \implies 25 = 121 - 8c \implies 8c = 96 \implies c = 12

Estimated Time:1m 30s

Question