SAT Nonlinear equations in one variable and systems of equations in two variables Practice Question 17 (Hard)

Question bank / Nonlinear equations in one variable and systems of equations in two variables

Math · Advanced MathHard

$y = -x^2 + 6x - 5$
$y = k$
In the system of equations above, $k$ is a constant. For which of the following values of $k$ does the system have no real solutions?

$0$

$2$

$4$

$5$

Show answer and explanation

Answer: D

The parabola opens downward, with vertex at $x = -\frac{6}{2(-1)} = 3$ and maximum $y = -9 + 18 - 5 = 4$ . A horizontal line $y = k$ misses the parabola when it lies above the maximum, so $k > 4$ . Only 5 works.

Choice C is the maximum value itself; the line $y = 4$ touches the vertex, giving exactly one solution.

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