According to the Rational Root Theorem, integer solutions of this equation are factors of the constant term - 4.
Let's list all possibilities.
± 1, ± 2, ± 4
We can use synthetic division to try these possibilities.
Let's start with x=1.
rl IR-0.15cm r 1 & |rr 1& -4& 6& -4 & 1& -3& 3 & c 1& -3 & 3 & -1
Since the last entry of the last line is -1, x=1 is not a solution.
Let's try x=-1 next.
rl IR-0.15cm r -1 & |rr 1& -4 & 6& -4 & -1 & 5& -11 & c 1& -5& 11& -15
Since the last entry of the last line is -15, x=-1 is not a solution.
Let's try x=2 next.
rl IR-0.15cm r 2 & |rr 1& -4& 6& -4 & 2& -4& 4 & c 1& -2 & 2 & 0
Since the last entry of the last line is 0, we know that x=2 is a solution. We also get the factor form of the expression.
x^3-4x^2+6x-4=(x-2)(x^2-2x+2)
To find the other solutions, we can set the quadratic factor equal to 0 and solve the equation.
x^2-2x+2=0
Let's check the discriminant first.
b^2-4ac=(-2)^2-4(1)(2)=-4
This is negative, so there are no real solutions.
The only solution of f(x)=g(x) is the one we found above using synthetic division.
Solution: x=2
Graphing Approach
Let's use a graphing calculator to check our answer.
We begin by pushing the Y= button and typing the equations in the first two rows.
To see the graphs, let's use the standard window.
Push WINDOW, check the settings, and push GRAPH.
Next, to find the intersection, push 2nd and TRACE. From this menu, choose intersect.
The calculator will prompt you to choose the first and second curve and to provide the calculator with a best guess of where the intersection might be.
We can see that the algebraic and graphing approach gave the same solution.
Solution: x=2
