We can solve any quadratic equation by completing the square. It is possible to use this method on a general quadratic equation to get a general formula. Let's give this a try. Recall that any quadratic equation can be written in standard form.

a x^{2} + b x + c = 0

In this form

a,

b,

and

c

are real numbers, and

a \neq = 0 .

Remember that we can complete the square of an expression

x^{2} + d x

by adding

(\frac{d}{2})^{2} .

Therefore, our next step is to take the left-hand side of the equation to this form.

a x^{2} + b x + c = 0

SubEqn

$LHS - c = RHS - c$

a x^{2} + b x = - c

DivEqn

$LHS / a = RHS / a$

x^{2} + \frac{b}{a} x = - \frac{c}{a}

Now, we can compare the left-hand side with

x^{2} + d x

to determine the constant that we need to add to complete the square.

x^{2} + d x x^{2} + \frac{b}{a} x

As we can see,

d = \frac{b}{a} .

With this in mind we can proceed to find

(\frac{d}{2})^{2} .

(\frac{d}{2})^{2}

Substitute

$d = \frac{b}{a}$

(\frac{\frac{b}{a}}{2})^{2}

DivFracSL

$\frac{a}{b} / c = \frac{a}{b \cdot c}$

(\frac{b}{2 a})^{2}

CalcPow

Calculate power

\frac{b ^{2}}{4 a ^{2}}

Now let's add

\frac{b ^{2}}{4 a ^{2}}

to both sides of

x^{2} + \frac{b}{a} x = - \frac{c}{a}

to complete the square on the left-hand side. After that, we can factor the perfect square trinomial as the square of a binomial. This will allow us to solve the equation by taking the square root of both sides to isolate

Exercise