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High School Precalculus: Help and Review32 chapters | 297 lessons

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After reading this how to lesson, you'll be well on your way to using the completing the square method of solving quadratic equations. Learn what numbers you need to focus on to find your missing square.

Say you are trying to find the solutions to this problem:

You would set your y equal to 0 and then proceed to find the solutions. If you aren't able to easily factor this quadratic equation, then you can use the method that is called **completing the square**. In this method, you manipulate your equation so you end up with one squared part that equals a number. This way, you can easily find your two solutions.

Here are the steps to solve a quadratic by completing the square.

Whenever a problem asks you to find the solutions or x-intercepts, it means that you need to set your equation equal to 0 (i.e. set y = 0).

You want just your variables on the left and your numbers on the right. In our example, this means we move the 8 over to the other side. We can do so by adding it to both sides since it is being subtracted. Remember, when moving terms from one side to the other, you always perform the opposite or inverse operation.

You want your squared term to be just that, your variable squared, with no other constants multiplying with it. In our example, our squared term is being multiplied by a 3, so we need to divide both sides by 3.

This step is a little bit tricky. You're going to take the coefficient of the x-term, then you're going to divide it by 2. Then you're going to square this number and add it to both sides. So, for our example, the x-term's coefficient is 4 / 3. Dividing it by 2, we get 4 / 6 or 2 / 3. Then we'll square the 2 / 3 and add that to both sides. This is what we get:

Now that you've figured out the square of the coefficient of the x-term, you can now convert your equation into squared form. You'll use what you found to be half of the x-term's coefficient. You'll also add your like terms together on the right side of the equation. For our problem, this is what we get:

The next step in solving your equation is to take the square root of both sides. Doing this will cancel out your square. This is what we get for our problem:

Remember that when you take the square root of a number, you'll have both a positive and a negative component.

Now that you've canceled out your square, you can now to ahead and solve for your variable. Since you have a positive and a negative part, you'll have two equations to solve for. For our problem, you have these two equations you need to solve, one for the positive part and one for the negative part:

To find your solutions, solve for your variable by isolating it. For our problem, we'll need to subtract our 2 / 3 from both sides to find our solutions.

After isolating our variable, this is what we get for our answers:

We subtracted our 2 / 3 from both sides and simplified as much as we can. For the second solution, we took the minus sign from both numbers and put it outside the fraction. So instead of having a negative square root of 28 minus a 2, we have the negative of the square root of 28 plus 2.

Let's look at another example.

Find the solutions to this equation:

To find the solutions to this equation, you first have to set the equation equal to 0. Then you need to move the constant number over to the other side.

Then you'll divide by the coefficient of the x squared term if there is one. In this case, you'll divide by 2 on both sides of the equation.

Now, you'll look at the coefficient of the x-term. It is -1. You'll divide this by 2 and then add the square of this to both sides of the equation.

In the next step, you now rewrite your equation in squared form and you add your like terms together. To write your equation in squared form, you'll take the coefficient of your x-term from the equation you got from the last step and you divide it by 2.

To solve the squared form, you now take the square root of both sides. Remember that when you take the square root, you'll always have two parts, one negative and one positive. Now, if at this point, you end up taking the square root of a negative number, then that means your equation has no solution.

Now you can go ahead and solve for your variable. Remember, you'll have two equations to solve for. To solve each equation, you'll need to add 1 / 4 to both sides of the equation. You get this for your two solutions.

And you are done!

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14 in chapter 15 of the course:

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High School Precalculus: Help and Review32 chapters | 297 lessons

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