How do you solve x^2 + 12x + 20 = 0x2+12x+20=0 by completing the square?

2 Answers
Mar 3, 2018

x=-10, x=-2x=10,x=2

Explanation:

Our Quadratic is in the form ax^2+bx+c=0ax2+bx+c=0. When we complete the square, we take half of the bb value, square it, and add it to both sides.

First, let's subtract 2020 from both sides. We get:

x^2+12x =-20x2+12x=20

Half of 1212 is 66, and when we square 66, we get 3636. Let's add this to both sides. We get:

x^2+12x+36=-20+36x2+12x+36=20+36

Simplifying, we get:

x^2+12x+36=16x2+12x+36=16

Now, we factor the left side of the equation. We think of two numbers that add up to 1212 and have a product of 3636. 66 and 66 are our two numbers.

We can now factor this as:

color(blue)((x+6)(x+6))=16(x+6)(x+6)=16

Notice, what I have in blue is the same as x^2+12x+36x2+12x+36. We can rewrite this as:

(x+6)^2=16(x+6)2=16

Taking the square root of both sides, we get:

x+6=-4x+6=4 and x+6=4x+6=4

Subtracting 66 from both sides of the equations, we get:

x=-10, x=-2x=10,x=2

Mar 3, 2018

x=-2x=2 and x=-10x=10

Explanation:

By completing the square, we always half the coefficient of xx, putting it in brackets and squaring it, as a quadratic is always in the form ax^2+bx+cax2+bx+c. Always remember to add the constant of (+20)(+20) on the end:

therefore x^2+12x -> (x+6)^2

Also by completing the square, we also take away the squared number of half of the previous coefficient (the number in the brackets).

(x+6)^2-36+20

Simplifying terms:

-36+20=-16

Plugging this back in, removing the -36 and 20:

(x+6)^2-16

This is in the completed square form. You can always check this by expanding out:

(x+6)(x+6) -> x^2+6x+6x+36-16 -> x^2+12x+20

therefore This complete the square form is correct.

Solving; so far we know:

(x+6)^2-16=0

So we add 16 to solve this, as we cannot solve when equal to 0.

(x+6)^2=16

As we do not want squares brackets, the opposite of squaring is square rooting, and this cancels out the squared brackets:

(x+6)^2=16 -> x+6=pmsqrt16

Always remember the pm when square rooting...

As we want x on its own, we have to -6 to get rid of it, transferring the -6 to the other side of the equation.

x+6=pmsqrt16 -> x=-6pmsqrt16

Always remember there are mostly two solutions, as there is a plus and minus on the answer.

therefore our answers are...

x=-6pmsqrt16 and x=-6pmsqrt16

But using our squared numbers:

1^2=1xx1=1
2^2=2xx2=4
3^2=3xx3=9
4^2=4xx4=16 sqrt16 can be simplified further as it is a squared number...

x=-6pmsqrt16 -> x=-6pm4

therefore the answers are:

x=-6+4 and x=-6-4 since most of the time we have two solutions.

These simplify to:

x=-2 and x=-10