Example of Completing the Square

x2+x+=0

Enter the 3 coefficients a,b,c of the quadratic equation in the above 3 boxes.
Next, press the button to find the solution with steps.

Completing the Square

Here is how to Solve by Completing the Square.
The Quadratic Equation in Standard Form is
$$\boxed{ y=x^2+bx+c }$$

To Solve by Completing the Square we add and subtract
$$\textcolor{blue}{({b \over 2})^2}$$ which yields:
$$\boxed{ y=x^2+bx+ \textcolor{blue}{({b \over 2})^2} + c – \textcolor{blue}{({b \over 2})^2} }$$

Completing the Square yields:
$$\boxed{ y=(x+b/2)^2 + c – ({b \over 2})^2 }$$

which is the Complete the Square Formula.

Another Example of Completing the Square

We are given the Quadratic Equation below in Standard Form
$$y=x^2- 6x-7$$ .

$$({ – b \over 2a})^2= ({ -(-6)\over (2*1)})^2 = 3^2=9$$ .

Thus, we have:
$$y=(x^2- 6x+ \textcolor{blue}{9})-7- \textcolor{blue}{9}$$

This allows us to Solve via Completing the Square:
$$y=(x-3)^2-16$$ .

To solve $$(x-3)^2-16 =0$$ we first add 16:

$$(x-3)^2=16$$ . Next, we take the Square Root:

$$x-3=\pm 4$$ . Adding 3 to $$\pm 4$$ yields the 2 solutions:

$$x=7 , x=-1$$ .

Easy, wasn’t it?

Tip: When using the Complete the Square Solver to solve
$$x^2-6x-7=0$$ we must enter the 3 coefficients as
$$a=1, b=-6$$ and $$c=-7$$.

Then, the Solver will first Complete the Square to find the Vertex $$(h,k)=(3,-16)$$ .

Thus, the Vertex Form of the Parabola is $$y=(x-3)^2-16$$ .

Solving $$(x-3)^2-16=0$$ yields the two zeros: $$x=7 , x=-1$$ .

Get it now? Try the above Complete the Square Solver again.