In the quadratic equation 2x^2 - 15x + c = 0, c is a constant. If one of the solutions to … — SAT

In the quadratic equation $2x^2 - 15x + c = 0$ , $c$ is a constant. If one of the solutions to the equation is $x = 6$ , what is the other solution?

Answer: 1.5

1.5

The sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-\frac{b}{a}

. For the equation

2x^2 - 15x + c = 0

, the sum of the roots is

-\frac{-15}{2} = 7.5

. Since one of the solutions is

6

, the other solution is

7.5 - 6 = 1.5

. Alternatively, substituting

x = 6

into the equation yields

2(6)^2 - 15(6) + c = 0 \implies 72 - 90 + c = 0 \implies c = 18

. The equation becomes

2x^2 - 15x + 18 = 0

, which factors as

(2x - 3)(x - 6) = 0

. Setting the factor

2x - 3 = 0

gives the other solution,

x = 1.5

(or

\frac{3}{2}

Find the sum of the roots of the quadratic equation using Vieta's formulas.

The sum of the roots is

-\frac{-15}{2} = 7.5

For any quadratic equation

ax^2 + bx + c = 0

, the sum of its roots is equal to

-\frac{b}{a}

Set up an equation using the known root to find the unknown root.

6 + x_2 = 7.5

, where

x_2

is the other root.

Since one root is

6

, adding it to the second root must equal the sum of the roots.

Solve for the second root.

x_2 = 1.5

Subtracting

6

from both sides of the equation yields

1.5

Sum of roots of a quadratic equation (Vieta's Formulas)

Estimated Time:1m 30s

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