A quadratic equation is an equation that can be written as

$$ax^2+bx+c=0,\qquad a\ne0.$$

Its solutions are the values of $x$ that make the quadratic polynomial equal to zero.

Solving by factoring

When the polynomial factors, the zero-product property gives the solutions: if $uv=0$, then $u=0$ or $v=0$.

For example,

$$x^2+5x+6=0$$

factors as

$$(x+2)(x+3)=0,$$

so

$$x=-2\qquad\text{or}\qquad x=-3.$$

Completing the square

A quadratic can also be rewritten as a square. For example,

$$x^2+6x+5=0$$

becomes

$$x^2+6x+9=4,$$

so

$$(x+3)^2=4.$$

Taking square roots gives $x+3=\pm2$, hence $x=-1$ or $x=-5$.

The quadratic formula

Completing the square in the general equation leads to the quadratic formula:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$

The expression

$$\Delta=b^2-4ac$$

is the discriminant. If $\Delta>0$, there are two distinct real solutions; if $\Delta=0$, there is one repeated real solution; and if $\Delta<0$, there are no real solutions.