In the quadratic equation 2x^2 - 11x + c = 0, c is a constant. If the sum of the squares o… — SAT

In the quadratic equation $2x^2 - 11x + c = 0$ , $c$ is a constant. If the sum of the squares of the two real solutions to the equation is $\frac{85}{4}$ , what is the value of $c$ ?

9Answer
B
18
C
-9
D
$\frac{9}{2}$

Answer

The sum of the solutions of the quadratic equation is

11/2

and the product of the solutions is

c/2

. The sum of the squares of the solutions is

(11/2)^2 - 2(c/2) = 121/4 - c

. Setting this equal to the given value of

85/4

and solving for

c

yields

c = 9

Step-by-Step Solution

Express the sum and product of the solutions using Vieta's formulas.

For the quadratic equation

2x^2 - 11x + c = 0

, the sum of the solutions is

x_1 + x_2 = -\frac{b}{a} = \frac{11}{2}

, and the product of the solutions is

x_1 x_2 = \frac{c}{a} = \frac{c}{2}

Vieta's formulas relate the coefficients of a polynomial to the sum and product of its roots.

Use the algebraic identity for the sum of squares of two variables.

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

This identity allows us to express the sum of squares in terms of the sum and product of the roots.

Substitute the expressions from Step 1 into the identity and set it equal to the given sum of squares.

\left(\frac{11}{2}\right)^2 - 2\left(\frac{c}{2}\right) = \frac{85}{4} \Rightarrow \frac{121}{4} - c = \frac{85}{4}

To solve for

c

using the given value of the sum of squares, which is

\frac{85}{4}

Isolate the constant

c

c = \frac{121}{4} - \frac{85}{4} = \frac{36}{4} = 9

To find the numerical value of

c

Key Concept

Vieta's Formulas and Root Identities

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic