Solve Quadratic Equations

When solving quadratic equations, the equation is set equal to zero, factorise, then each bracket is solved against zero.

Solve by factorising

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

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

Then, solve each factor:

$$x+2=0\text{and}x+3=0$$

$$x=-2\text{and}x=-3$$

Quadratics can either have 2, 1 or no solutions.

Using the Quadratic Formula

For any quadratic equation $a{x}^2+bx+c=0$, the quadratic formula is:

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

We can apply this formulae to any quadratic.

$$2x^2+3x-2=0$$

$$x=\frac{-3\pm \sqrt{3^2-4(2)(-2)}}{2(2)}$$

$$=\frac{-3\pm \sqrt{9+16}}{4}$$

$$=\frac{-3\pm \sqrt{25}}{4}$$

$$=\frac{-3\pm 5}{4}$$

So the solutions are:

$$x=\frac{2}{4}=\frac{1}{2}\text{or}x=\frac{-8}{4}=-2$$