Advanced Algebra 1Completing the Square

Advanced Algebra 1Completing the Square

Advanced Algebra 1Completing the Square


Another method of using factoring to solve quadratic equations is called the “completing the square” method.

In this method, the idea is to create the same factor TWICE (a perfect square), so that you can take the square root of both

sides in order to solve the quadratic equation.

In order to “complete the square”, you will take half of the middle (linear) term and square it to create a “perfect square”.

The reason for this is you need two numbers that add up to the middle term that are the same, and so the product must be

the same number times itself (which is the number squared).

Examples: What number would complete the square in the following situations? Then, list the factors.

x2 + 10x + ______x2 + 6x + ______x2 – 20x + ______

The key to this method is: WHATEVER YOU DO ON ONE SIDE OF THE EQUATION, YOU MUST DO ON THE OTHER SIDE. That means, if I add 16 on one side of the equation to complete the square, I must add 16 to the other side of the equation to keep it balanced.

You then solve the equation by taking the square root of both sides to isolate the variable.

EXAMPLESolve x2 – 6x – 15 = 0

If we tried to factor this, you would find that there are no numbers that add up to –6 and multiply to –15, so this equation is NOT FACTORABLE, and we wouldn’t get anywhere.

To solve it, we are going to complete the square.

Step 1: Clear some space so that you can complete the square. In this case, we want to add 15 to the other side.

Step 2: Determine what number will complete the square on the left side. Here, the correct answer would be 9.

(Half of –6 and square it so it becomes a perfect square)

Reason: x2 – 6x + 9 = (x – 3)(x – 3) = (x – 3)2

Step 3:Balance both sides of the equation. Since we added 9 on the left side to complete the square, we need to add 9 to the right side.

(x2 – 6x + 9) = 15 + 9

Step 4:Factor the left side into the square of a binomial (meaning the same term twice).

(x – 3)(x – 3) = 24

(x – 3)2 = 24

Step 5:Take the square root of both sides. Remember to use when taking the square root of a number.

x – 3 = ±

Step 6:Solve for x by completely isolating the variable. Here, you are adding 3 to both sides. Remember, DO NOT add a radical and a non-radical as they are not like terms. Please also put the radical last.

x = 3 ± NOT TERRIBLY DIFFICULT!!! And we didn’t even need a calculator

You should be able to find the EXACT SOLUTIONS (leaving square roots in your answer), as well as APPROXIMATE SOLUTIONS (using your calculator and rounding to the appropriate number of decimals places.)


1.x2 – 8x – 3 = 02.x2 + 2x = 13

EXAMPLE ONE: Solve the equation by finding square roots.

1) x² + 8x + 16 = 92) x² - 6x + 9 = 253) x² - 12x + 36 = 49

X: ______X: ______X: ______

EXAMPLE TWO: Making a perfect square trinomial

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as a square of a binomial.

1) x² - 10x + c2) x² + 12x + c3) x² - 40x + c

C: ______C: ______C: ______


4) x² - 18x + c5) x² + 5x + c6) x² - x + c

C: ______C: ______C: ______


EXAMPLE THREE: Solving the equation by completing the square

1) x² - 10x + 10 = 02) x² + 8x – 1= 03) x² - 14x + 10 = 0

Try these more difficult ones.

4) 5x² – 10x + 30 = 05) 6x² – 12x – 18 = 0

Finally, to convert an equation from STANDARD FORM into VERTEX FORM, use completing the square.

EXAMPLE FOUR: Write the quadratic function in vertex form. Then identify the coordinates of the vertex.

1) y = x² - 12x + 382) y = x² - 14x + 50

Form: ______Form: ______

Vertex: ______Vertex: ______