Solving Quadratic equations by completing the squares - SS2 Mathematics Lesson Note

Some quadratic equations cannot be solved by factorisation. These equations can be solved via the completing the square method which involving making the equation a perfect square that can be factorised.

Example 1 Solve the equation ${2x}^2+3x-2=0$ using the completing the square method.

Solution

STEP 1: make the coefficient of $x^2$ unity by dividing every term by $2$

$x^2+\frac{3}{2}x-1=0$

STEP 2: transfer any constant term to the RHS of the equality sign

$x^2+\frac{3}{2}x=1$

STEP 3: make the LHS a perfect square by adding the square of half the coefficient of $x$ to both sides

$x^2+\frac{3}{2}x+{(\frac{3}{4})}^2=1+{(\frac{3}{4})}^2$

$x^2+\frac{3}{2}x+\frac{9}{16}=1+\frac{9}{16}$

STEP 4: factorise the LHS

${(x+\frac{3}{4})}^2=\frac{25}{16}$

$x+\frac{3}{4}=\sqrt{\frac{25}{16}}$

$x+\frac{3}{4}=\pm \frac{5}{4}$

$x=-\frac{3}{4}\pm \frac{5}{4}=-\frac{3}{4}+\frac{5}{4}or-\frac{3}{4}-\frac{5}{4}$

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

$x=\frac{1}{2}or-2$

From this method, we obtain the much easier to use quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. Here, $b^2-4ac$ is called the discriminant denoted in math by the letter D and is used to determine what nature the roots possess. Given an equation, ${ax}^2+bx+c=0$:

1. If $b^2-4ac>0$, then the roots are real and distinct

2. If $b^2-4ac<0$, then the roots are imaginary or complex (usually involving the square of a negative number)

3. If $b^2-4ac=0$, then the roots are real and equal (coincidental)