Quadratic Equations

Quadratic equations can be solved by factorising, completing the square and using a formula. In this section you will learn how to:

solve quadratic equations by factorising
solve quadratic equations by completing the square
solve quadratic equations by using the formula
solve simultaneous equations when one of them is quadratic

Solving quadratic equations by factorising

Unless a graphical method is asked for, quadratic equations on the non-calculator paper will probably involve factorising or completion of the square. Quadratic equations can have two different solutions or roots.

You may need a quick look at 'factorising' again to remind yourself how to factorise expressions such as:

$x2 − x − 6$

which factorises into $(x − 3)(x + 2)$,

$a^2 − 3a$

which factorises into $a(a − 3)$

and

$b^2 − 2b + 1$

which will factorise into $(b − 1)^2$.

This video shows you how to solve a quadratic equation by factoring.

If two numbers or expressions are multiplied and the result is zero, then one or both of them must be zero.

If $A x B = 0$ then $A = 0$ OR $B = 0$.

If $(x – 3)(x + 2)=0$ then $(x – 3)= 0$ or $(x + 2)=0$

And if $(x - 3) = 0$ then $x = 3$

Or if $(x + 2) = 0$ then $x = -2$

Solving quadratic equations by completing the square

Example:

$(x + 3)^2 – 4 = 0$

Add 4 to each side

$(x + 3)^2 =4$

$(x + 3) = \pm 2$

Take the square root of each side, $\sqrt{4} \pm 2 (-2 \times 2 = 4, 2 \times 2 = 4)$

So $x = -3 \pm 2$

$x = -1$ and $x = -5$

NOTE: Check by substituting both roots back into the original equation.

Completing the Square involves rearranging the Quadratic Equation into the form

$$(x + a)^2 – b = 0$$

Where $a$ and $b$ are numbers so that

$$(x + a)^2 = b$$

Taking the Square Root of both sides gives

$$(x + a) = \pm \sqrt{b}$$

Giving $x = \pm \sqrt{b} – a ( \pm \sqrt{b}$ gives two roots)

Completing the Square can be used to give answers to a given accuracy or in Surd form (eg $x = \sqrt{5}, x = \sqrt{3}$)

Solving quadratic equations by using the formula

When using the quadratic formula, don’t forget the ‘2a’denominator. Also, be careful when dealing with negative numbers

inside the square root. State your values of a,b and c to be used in the formula.

NOTE: The above calculations are easily checked − especially if your calculator can store numbers as variables.

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