If x^2 - 9x + 18 = 0, and x_1 and x_2 are the real roots of the equation such that x_1 > x… — GMAT

If $x^2 - 9x + 18 = 0$ , and $x_1$ and $x_2$ are the real roots of the equation such that $x_1 > x_2$ , what is the value of $2x_1 - x_2$ ?

Answer: 9

The value of

2x_1 - x_2

is 9.

Factoring the quadratic equation

x^2 - 9x + 18 = 0

yields

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

. The roots are

x = 6

and

x = 3

. Given that

x_1 > x_2

, we assign

x_1 = 6

and

x_2 = 3

. Evaluating

2x_1 - x_2

gives

2(6) - 3 = 9

Factor the quadratic expression

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

Find two numbers that multiply to

18

and add to

-9

, which are

-6

and

-3

Solve for the roots of the equation

x = 6

x = 3

Set each factor equal to zero:

x - 6 = 0 \implies x = 6

and

x - 3 = 0 \implies x = 3

Assign root values based on the condition

x_1 > x_2

x_1 = 6

and

x_2 = 3

Since

6 > 3

, the larger root is assigned to

x_1

and the smaller root to

x_2

Evaluate the target expression

Substitute

x_1 = 6

and

x_2 = 3

into

2x_1 - x_2 = 2(6) - 3 = 12 - 3 = 9

Solving Quadratic Equations by Factoring

Question