Question 307529
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:


{{{graph(600,600,-10,10,-10,10,x^2 - x - 10)}}}


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