In the system of equations below, k is a constant. \begin{aligned} y &= 3x^2 - 12x + 7 \\ … — SAT

In the system of equations below, $k$ is a constant.

\begin{aligned} y &= 3x^2 - 12x + 7 \\ y &= kx - 5 \end{aligned}

If the system has two distinct real solutions, $(x_1, y_1)$ and $(x_2, y_2)$ , such that the average of the $x$ -coordinates of the solutions is $3$ , what is the value of $k$ ?

A
-6
B
-3
6Answer
D
0

Answer

To find the value of

k

, we set the two equations equal to each other to solve for the

x

-coordinates of the intersection points:

3x^2 - 12x + 7 = kx - 5

. Moving all terms to one side gives the quadratic equation

3x^2 - (12+k)x + 12 = 0

. The sum of the roots

x_1 + x_2

is given by

-\frac{b}{a} = \frac{12+k}{3}

. The average of the roots is

\frac{x_1 + x_2}{2} = \frac{12+k}{6}

. Since the average of the

x

-coordinates is

3

, we set

\frac{12+k}{6} = 3

, which gives

12+k = 18

, and solving for

k

yields

6

. Evaluating the discriminant with

k = 6

gives

(-18)^2 - 4(3)(12) = 324 - 144 = 180 > 0

, confirming two distinct real solutions.

Step-by-Step Solution

Equate the two equations to find the

x

-coordinates of the intersection points.

3x^2 - 12x + 7 = kx - 5

The intersection points of the system occur where the

y

-values of the two functions are equal.

Rearrange the equation into standard quadratic form:

ax^2 + bx + c = 0

3x^2 - (12+k)x + 12 = 0

Grouping the like terms allows us to identify the quadratic coefficients

a = 3

b = -(12+k)

, and

c = 12

Express the sum and the average of the solutions in terms of

k

using Vieta's formulas.

Sum of solutions

x_1 + x_2 = -\frac{b}{a} = \frac{12+k}{3}

, so the average is

\frac{x_1 + x_2}{2} = \frac{12+k}{6}

Vieta's formulas state that the sum of the roots of a quadratic equation is

-\frac{b}{a}

. The average is half of the sum.

Set the average equal to

3

and solve for

k

\frac{12+k}{6} = 3 \implies 12+k = 18 \implies k = 6

The problem states that the average of the

x

-coordinates of the solutions is

3

Verify that

k = 6

yields two distinct real solutions by checking the discriminant.

For

k = 6

, the equation is

3x^2 - 18x + 12 = 0

. The discriminant is

D = (-18)^2 - 4(3)(12) = 324 - 144 = 180

. Since

D > 0

, two distinct real solutions exist.

A positive discriminant ensures that there are two distinct real solutions, satisfying the problem's condition.

Key Concept

Solving nonlinear systems of equations using quadratic properties and Vieta's formulas.

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