A system of equations consists of the equations y = -x^2 + 2x + 7 and y = 6x + k, where k … — SAT

A system of equations consists of the equations $y = -x^2 + 2x + 7$ and $y = 6x + k$ , where $k$ is a constant. If the system has two distinct real solutions, what is the greatest integer value of $k$ ?

Answer: 10

Answer

The correct answer is 10. The greatest integer value of the constant that allows the system to have two distinct real solutions is 10.

To find the number of solutions to the system, equate the two equations:

-x^2 + 2x + 7 = 6x + k

. Rearranging this equation into standard quadratic form gives

x^2 + 4x + (k - 7) = 0

. For the system to have two distinct real solutions, the discriminant of this quadratic equation must be strictly greater than zero. The discriminant is calculated as

b^2 - 4ac = 4^2 - 4(1)(k - 7) = 16 - 4k + 28 = 44 - 4k

. Setting this greater than zero yields

44 - 4k > 0

, which simplifies to

k < 11

. The greatest integer value of

k

that is strictly less than 11 is 10.

Step-by-Step Solution

Equate the expressions for

y

from both equations.

-x^2 + 2x + 7 = 6x + k

At the points of intersection, the

y

-values of both equations must be equal.

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

x^2 + 4x + (k - 7) = 0

Standard form is required to calculate the discriminant of the quadratic equation.

Write the expression for the discriminant

\Delta = b^2 - 4ac

using the coefficients from the quadratic equation.

\Delta = 4^2 - 4(1)(k - 7) = 44 - 4k

The discriminant determines the number of real solutions to the quadratic equation.

Set the discriminant to be strictly greater than 0 and solve the inequality for

k

44 - 4k > 0 \implies k < 11

For the system to have two distinct real solutions, the discriminant must be positive.

Determine the greatest integer value of

k

that satisfies the inequality

k < 11

The largest integer strictly less than 11 is 10.

Key Concept

Using the discriminant of a quadratic equation derived from a nonlinear system to determine the number of real solutions.

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