Solving an equation involving $x^2$

Question

I have come to a question with the equation:

$$6 = x^2 -7x + 6.$$

The answer is $7$.

How do I do I find the solution to a problem involving $x^2$?

Answer 1

Hint: The $6$'s cancel out, then factoring out $x$ leaves you with $$x^2-7x=0 \implies x(x-7)=0$$

Answer 2

Subtracting six from both sides leaves $$x^2 - 7x = 0.$$ Now note that both terms on the left hand side have contain $x$ so we can take it out as a common factor like this: $$x(x - 7) = 0.$$ Now, we use the null factor law that tells us that if the product of two things is zero, at least one of them is zero. Therefore we must have $x = 0$ or $x - 7 = 0$ (i.e. $x = 7$).

So the solutions to $x^2 - 7x + 6 = 6$ are $x = 0$ and $x = 7$.

Answer 3

If you ever get stuck and can't apply the methods here, there is a more general solution. If

$$Ax^2+Bx+C=0$$ then $$x=\frac{-B \pm \sqrt{B^2-4AC}}{2A}$$

In this case

$$x^2 - 7x=0$$ implies that

$$x=\frac{7 \pm \sqrt{49-4(1)(0)}}{2(1)}=\frac{7 \pm \sqrt{49}}{2}=\frac{7 \pm 7}{2}=$$

which equals $0$ or $(7+7)/2=7$