Question

Solve the quadratic equation by completing the square.

x^2+6x-6=0

First, choose the appropriate form and fill in the blanks with the correct numbers.
Then, solve the equation. If there is more than one solution, separate them with commas.

Answer 1

For completing squares first take the constant to the other side of the equal sign.

\[x ^{2} + 6x = 6\] Now we have to complete the perfect square trinomial (PST) on the left hand side.

You know the PST is of the form

\[a ^{2} \pm 2ab + b ^{2}\]

Whose factors will be \[(a + b)^{2} \space or \space (a - b )^{2}\] depending on + or _ in the second term.

so if you see the middle term 2ab it is 2 times a times b

So in our problem 2ab = 6x

Our a = x and b is what we are trying to find.

we know 2ab = 6x

i.e 2 x b = 6x

so b = 3

Now the last term in the PST is b^2 = 9

So our PST on the left will be \[x ^{2} + 6x + 9\] Since we added 9 to the left we need to add 9 to the right to make it balanced.

So your equation will be

\[x ^{2} + 6x + 9 = 6 + 9 = 15\] \[x ^{2} + 6x + 9 = 15\]

Answer 2

Now since the left is a PST we can write the factors as (a + b)^2

a = x and b = 3

\[( x + 3)^{2} = 15\]

Answer 3

so you mean x= sqrt(15)-3, -sqrt(15)-3