Solve the equation for all values of x by completing the square

Expert intro: Completing the square turns the quadratic into a perfect square trinomial, making the roots easy to find.

Detailed walkthrough

We start with

$x^2-6x=-8$

Step 1: Complete the square

Take half of $-6$, which is $-3$, and square it:

$(-3)^2=9$

Add 9 to both sides:

$x^2-6x+9=-8+9$

Step 2: Rewrite the left side

$(x-3)^2=1$

Step 3: Take square roots

$x-3=\pm 1$

So:

$x=3\pm 1$

That gives

$x=4 \quad \text{or} \quad x=2$

Answer

$\boxed{x=2,4}$

💡 Pitfall guide

Do not add 3 instead of 9 when completing the square; you must square half of the coefficient of $x$. Also, remember to take both the positive and negative square roots after isolating the square.

🔄 Real-world variant

If the equation were $x^2-6x=-9$, then adding 9 would give $(x-3)^2=0$, so the only solution would be $x=3$. The method is the same, but the final number on the right changes the number of solutions.

🔍 Related terms

completing the square, perfect square trinomial, quadratic formula