If one of the solutions to the equation 2x^2 - 9x + c = 0, where c is a constant, is x = 2… — SAT

If one of the solutions to the equation $2x^2 - 9x + c = 0$ , where $c$ is a constant, is $x = 2$ , what is the other solution to the equation?

A
$-\frac{13}{2}$
B
$-\frac{5}{2}$
$\frac{5}{2}$ Answer
D
$7$

Answer

The other solution to the equation is

\frac{5}{2}

The correct answer is found by using the relationship between the coefficients of a quadratic equation and the sum of its roots. For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

. Substituting the values

a = 2

and

b = -9

, we find that the sum of the roots is

-\frac{-9}{2} = \frac{9}{2}

. Since one of the solutions is given as

2

, the other solution must be

\frac{9}{2} - 2 = \frac{5}{2}

. Alternatively, one can find the constant

c

by substituting

x = 2

into the equation to get

2(2)^2 - 9(2) + c = 0

, which yields

c = 10

. Solving the resulting equation

2x^2 - 9x + 10 = 0

by factoring gives

(2x - 5)(x - 2) = 0

, which confirms the other solution is

\frac{5}{2}

Step-by-Step Solution

Identify the relationship between the coefficients and the sum of the roots for a quadratic equation.

For any quadratic equation in the form

ax^2 + bx + c = 0

, the sum of its roots,

x_1

and

x_2

, is given by the formula

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

This relationship allows us to find the second root directly without needing to calculate the value of the constant

c

first.

Substitute the given root and the coefficients of the quadratic equation into the sum of the roots formula.

Here,

a = 2

and

b = -9

. The given root is

x_1 = 2

. Substituting these values gives

2 + x_2 = -\frac{-9}{2}

, which simplifies to

2 + x_2 = \frac{9}{2}

This sets up a linear equation to solve for the unknown root,

x_2

Solve for the second root

x_2

x_2 = \frac{9}{2} - 2 = \frac{9}{2} - \frac{4}{2} = \frac{5}{2}

This isolates the unknown variable to find the final value.

Key Concept

Using the sum of roots formula (

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

) to solve for an unknown solution of a quadratic equation.

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