For a certain constant k, the quadratic equation x^2 - 12x + k = 0 has two real roots, r_1… — GMAT

For a certain constant $k$ , the quadratic equation $x^2 - 12x + k = 0$ has two real roots, $r_1$ and $r_2$ . If $r_1^2 + r_2^2 = 94$ , what is the value of $k$ ?

Answer: 25

Answer

Applying Vieta's formulas gives

r_1 + r_2 = 12

and

r_1 r_2 = k

. Substituting these into the identity

(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2

yields

12^2 = 94 + 2k

. Solving

144 = 94 + 2k

gives

2k = 50

, so

k = 25

Step-by-Step Solution

Apply Vieta's formulas to the given quadratic equation

x^2 - 12x + k = 0

The sum of the roots is

r_1 + r_2 = 12

and the product of the roots is

r_1 r_2 = k

For a quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and product is

c/a

Expand the square of the sum of the roots

(r_1 + r_2)^2

(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2

This algebraic identity connects the sum of roots, sum of squared roots, and product of roots.

Substitute

r_1 + r_2 = 12

r_1^2 + r_2^2 = 94

, and

r_1 r_2 = k

into the identity.

12^2 = 94 + 2k \implies 144 = 94 + 2k

Replacing terms with known numerical values creates a linear equation in

k

Isolate and solve for

k

2k = 144 - 94 = 50 \implies k = 25

Basic algebraic manipulation yields the exact value of

k

Key Concept

Vieta's Formulas and Symmetric Polynomial Identities

Estimated Time:1m 30s

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