Question

Two functions are shown in the table below.

Function: 1 2 3 4 5 6
f(x) = -x^2 + 4x+12

g(x) = -x + 6

Complete the table on your own paper, then select the value that is a solution to f(x) = g(x).

Answers:
A) x= 2
B) x=3
C) x=5
D) x=6

Answer 1

To find which x coordinate makes f(x) = g(x) or to find when those lines intersect, we need to set them equal: $- {x}^{2} + 4x + 12 = - x + 6$

Add x-6 to both sides of the equation: $- {x}^{2} + 5x + 6 = 0$

Make $- {x}^{2}$ positive by multiplying the equation by -1: ${x}^{2} - 5x -6 = 0$

Now find the x using quadratic formula: $x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a}$

Plug the values: $x = \frac{ - ( - 5)\pm \sqrt{ { (- 5)}^{2} - 4 \times 1 \times - 6 } }{2 \times 1}$

Simplify:
$x = \frac{5 \pm \sqrt{25 + 24} }{2} = \frac{5\pm \sqrt{ 49} }{2}$

Now we need to branch the plus-minus sign:
$x = \frac{5 + 7}{2} = \frac{12}{2} =6$
$x = \frac{5 - 7}{2} = \frac{ - 2}{2} = - 1$

Because -1 isn't in the choices, 6 is the right answer.