Question

The parabola $y=x^2+5x+6$ is intersected by the line $y=-\frac{1}{2}x+12$. What is the y-coordinate of the intersection closest to the x-axis?

• A. $10$
• B. $12$
• C. $4$
• D. $2$

Answer 1

The correct option is A $10$

To solve the problem without a graphing calculator, set the two equations equal to each other:

$-x^2+5x+6=x+12$.

Multiply the entire equation by 2 to get

$-2x^2+10x+12=\pm 24$

Rewrite the equation so that it equals 0, so it becomes

$-2x^2+11x-12=0$.

Multiply the entire equation by -1 to get,

$2x^2-11x+12=0$

Factor the quadratic to get,

$(2x-3)(x-4)=0$

Solve for the two possible values of $x$:

If $2x-3=0,$ then $x=\frac{3}{2}$ and if $x-4=0,$ then $x=4$.

Because the slope of the line is negative, the x-value of the point that is farthest to the right along the x-axis must also be closer to the x-axis.

Plug $4$ in for $x$ in the second equation to get $y=-\frac{1}{2}\left(4\right)+12=10$.

Hence, the correct answer is $10$.