Solving quadratic equations by completing the square

How to complete the square

For most quadratic equations, you’ll have to use the method of “completing the square” to convert an equation so that x equals a number.

When moving the quadratic equation around to carry out completing the square, be careful with the signs on the x-terms when you move them over. If you forget to change a sign from positive to negative, or vice versa, you’ll end up getting the wrong answer. Also make sure that you’re not leaving out the plus-or-minus sign after square rooting, or that you’re forgetting to draw in the square root sign altogether.

Solving quadratic equations by completing the square

Now that we know the basics of what completing the square is, let’s look at an example and try the concept out.

completing the square step 1
completing the square step 1

How do we solve 2x^2-12x+10=0?

completing the square step 2
completing the square step 2

Solving quadratic equations with completing the square requires us to carry out a series of steps. You can say that there’s a completing the square formula.

First, we’ll move all the x’s to one side of the equation. We can see that in this question, we have two terms with x’s in them. Let’s first move the 10 over to the right hand side by subtracting both sides by 10. This will cancel out the 10 on the left-hand-side, and leave a -10 on the right hand side.

For the second step, we’ll factor out the leading coefficient of the x^2 term. In front of the x^2 term, we see that our coefficient is 2. If we factor it out, it’ll leave us with:

completing the square step 3
completing the square step 3

We’ll get 2(x^2-6x)=-10. The -10 on the right remains untouched from the previous step.

Now it’s time for us to carry out completing the square. We’re going to add equal values to both sides of the equation.

completing the square step 4
completing the square step 4

So what value are we going to add inside the brackets in the above? It’s actually determined by the coefficient of the x-term. We’ll take the coefficient and divide it by 2. Therefore, we’ll get -6 divided by 2, which gives us -3.

completing the square step 5
completing the square step 5

Remember to have -3 on both the left and right side of the equation to balance it out.

For the final step to doing completing the square, it’s mainly cleaning up our work. We can first convert the numbers inside the bracket on the left side to a perfect square. But what perfect square are we converting it to? We can see from our work that we have a x^2 and also a (-3)^2. These both have squares. This gives us 2(x-3)^2.

To clean up the right side of the equation, we’ll get positive 9 from (-3)^2. Calculate the numbers together on the right hand side and you’ll get a total of 8.

completing the square step 6
completing the square step 6

Now we are ready to solve for x. Let’s divide both sides of the equation we have after cleaning up by 2. This will cancel out the 2 on the left, and also simplify the right to 4.

completing the square step 7
completing the square step 7

Then to remove the square on the left, take the square root of both sides. When we try to solve for an unknown in an equation, and we take a square root, always remember that in front of the square root there’s always a plus and minus sign. Next, to solve for x, all we have to do is get rid of the -3 on the left side by adding 3 to both sides of the equation. This will give us our final answer:

completing the square step 8
completing the square step 8

X is equaled to either 5 or 1.

These are completing the square steps to using completing the square to solve for an unknown in a quadratic equation! For an intro video to recap the concept from here. If you needed another step-by-step walk through, try out this one!

Solving quadratic equations by completing the square

When a quadratic equation cannot be factorized, we can use the method of completing the square to solve the equation.

Lessons

Notes:
4-step approach:
1. isolate X’s on one side of the equation
2. factor out the leading coefficient of X2X^2
3. “completing the square”
• X-side: inside the bracket, add (half of the coefficient of X)2X)^2
• Y-side: add [ leading coefficient \cdot (half of the coefficient of X)2X)^2 ]
4. clean up
• X-side: convert to perfect-square form
• Y-side: clean up the algebra
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Solving quadratic equations by completing the square

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