SOLUTION: f(x)= x^2-6 and g(x)= x+4, for what values of x does f(gx)= g(x)?

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Question 307529: f(x)= x^2-6 and g(x)= x+4, for what values of x does f(gx)= g(x)?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
Original equations are:

f(x) = x^2 - 6
g(x) = x + 4

f(x) = g(x) means:

x^2 - 6 = x + 4

Subtract x from both sides of the equation and subtract 4 from both sides of the equation to get:

x^2 - x - 10 = 0

A graph of this equation looks like this:

The roots of this equation will be somewhere around -3 and 4.

They are not integers so you will need the quadratic formula to solve for them.

Equation is x^2 - x - 10 = 0
Standard form of the equation is ax^2 + bx + c = 0
a = 1
b = -1
c = -10

-b = 1
b^2 - 4ac = 1 - 4*1*-10 = 1 + 40 = 41
sqrt(b^2 - 4ac) = sqrt(41) = 6.403124237
2a = 2

x = (-b +/- sqrt(b^2-4ac))/(2a) becomes:

x = (1 +/- 6.403124235)/2 which becomes:

x = 3.701562119 and x = -2.701562119

Those are the roots of the quadratic equation we found using the quadratic formula.

The question was for what values of x does f(x) = g(x).

It should be these values.

We confirm by plugging these values into the original equation to get:

Original equation is f(x) = g(x) which became:

x^2 - 6 = x + 4

When x = 3.701562119, our equation becomes:

(3.701562119)^2 - 6 = 3.701562119 + 4 which becomes:

7.701562119 = 7.701562119 which is true, confirming that x = 3.701562119 is a good value for this equation.

When x = -2.701562119, our equation becomes:

(-2.701562119)^2 - 6 = (-2.701562119) + 4 which becomes:

1.298437881 = 1.298437881 which is true, confirming that x = -2.701562119 is a good value for this equation.

Both values are good so both values are your answer.

The values of x that make f(x) = g(x) are:

x = 3.701562119 and x = -2.701562119