# Given f(x)= x^2+7x+8, find x when f(x)= -4.

My answer is -4.

Find x if f(x)=0 and f(x)= 6x^2-7x-3.
My answer is -3.

Find the equation of the axis of symmetry of f(x)=x^2-12x+17.
I have no idea how to figure this problem.

## yes, -4 works but it is a parabola, two solutions

x^2 +7 x +8 = -4
x^2 + 7 x +12 = 0
(x+4)(x+3) = 0
x = -4 or x = -3

No, I did (2x-3)(3x+1) = 0
x = 3/2 or x = -1/3

y = x^2 -12 x +17
x^2 - 12 x = y-17
complete the square to look for symmetry
x^2 -12 x + 36 = y -17 +36
(x-6)^2 = y + 19
symmetry about x = 6
for example if x =8
(x-6)^2 = 2^2 = 4
and if x = 4, on the other side of x = 6 then
(x-6)^2 = (-2)^2 = 4 ,the same

## The axis of symmetry is a straight line which runs through the vertex of a parabola, cutting in in "half", and becoming a line of reflection.

Find the vertex by whatever method you learned, for your case you should get (6,-19)
then the equation of the axis of symmetry would be
x = 6

## To find the value of x when f(x) equals a specific value, you need to set the equation f(x) equal to that value and solve for x.

1. For f(x) = -4, we have the equation:
x^2 + 7x + 8 = -4

To solve this quadratic equation, you can rearrange it to have all terms on one side:
x^2 + 7x + 12 = 0

Now, factorize or use the quadratic formula to find the roots. In this case, you can factorize the equation as follows:
(x + 3)(x + 4) = 0

Setting each factor equal to zero gives:
x + 3 = 0 or x + 4 = 0

Solving for x in each equation yields:
x = -3 or x = -4

Therefore, the solutions to f(x) = -4 are x = -3 and x = -4.

2. For f(x) = 0, we have the equation:
6x^2 - 7x - 3 = 0

Similar to the previous example, you can factorize or use the quadratic formula to solve this equation. However, in this case, factoring might not be as straightforward. So, let's use the quadratic formula instead:
x = (-b ± √(b^2 - 4ac)) / (2a)

By substituting the values from the equation, we get:
x = (-(-7) ± √((-7)^2 - 4(6)(-3))) / (2(6))
x = (7 ± √(49 + 72)) / 12
x = (7 ± √121) / 12
x = (7 ± 11) / 12

This gives us two solutions:
x = (7 + 11) / 12 or x = (7 - 11) / 12

Solving further gives:
x = 18 / 12 or x = -4 / 12
x = 3/2 or x = -1/3

Hence, the solutions to f(x) = 0 are x = 3/2 and x = -1/3.

3. The equation of the axis of symmetry can be found by completing the square. The vertex form of a quadratic equation is f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.

In this case, the equation f(x) = x^2 - 12x + 17 is already in standard form, so we can proceed with completing the square:
f(x) = (x - 6)^2 - 36 + 17
f(x) = (x - 6)^2 - 19

From the vertex form equation, we identify that the axis of symmetry is given by x = h, which represents the x-coordinate of the vertex. In this case, the value of h is 6.

Therefore, the equation of the axis of symmetry of f(x) = x^2 - 12x + 17 is x = 6.