How to Solve Quadratic Equations: 3 Methods with Worked Examples

A quadratic equation has the form

$$a{x}^2+bx+c=0,a\ne 0$$ax^2 + bx + c = 0, a ≠ 0

There are three standard methods. Knowing which one to reach for saves time in exams.

1. Factorisation (splitting the middle term)

Use it when the roots are whole numbers or simple fractions.

Example: solve $2x^2-7x+3=0$2x^2 - 7x + 3 = 0.

1. Multiply $a$a and $c$c: $2\times 3=6$2 × 3 = 6.
2. Find two numbers that multiply to 6 and add to −7: they are −6 and −1.
3. Split the middle term: $2x^2-6x-x+3=0$2x^2 - 6x - x + 3 = 0.
4. Group: $2x(x-3)-1(x-3)=0\Rightarrow (2x-1)(x-3)=0$2x(x - 3) - 1(x - 3) = 0 ⇒ (2x - 1)(x - 3) = 0.
5. So $x=\frac{1}{2}$x = 1/2 or $x=3$x = 3.

2. Completing the square

Use it when you also need the vertex of the parabola, or when the question asks for this method.

Example: solve $x^2+6x+2=0$x^2 + 6x + 2 = 0.

$$x^2+6x=-2\Rightarrow x^2+6x+9=7\Rightarrow (x+3{)}^2=7$$x^2 + 6x = -2 ⇒ x^2 + 6x + 9 = 7 ⇒ (x + 3)^2 = 7 $$x=-3\pm \sqrt{7}$$x = -3 ± √(7)

3. The quadratic formula (Sridharacharya formula)

Works for every quadratic. In India it is taught as the Sridharacharya formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$x = frac -b ± sqrt b^2 - 4ac2a

The discriminant $\Delta =b^2-4ac$Δ = b^2 - 4ac tells you what kind of roots to expect before you finish:

• $\Delta >0$Δ > 0: two different real roots
• $\Delta =0$Δ = 0: one repeated real root
• $\Delta <0$Δ < 0: no real roots (two complex roots)

Example: $x^2+4x+5=0$x^2 + 4x + 5 = 0 has $\Delta =16-20=-4$Δ = 16 - 20 = -4, so

$$x=\frac{-4\pm \sqrt{-4}}{2}=-2\pm i$$x = (-4 ± √(-4))/(2) = -2 ± i

Which method should I use?

• Factorisation if you can spot the factors in under 30 seconds.
• Completing the square if you need the vertex or the question asks for it.
• The formula for everything else — it never fails.

Check your answer

Substitute each root back into the original equation. For $x=3$x = 3 in the first example: $2(9)-21+3=0$2(9) - 21 + 3 = 0 ✓.

Try your own equation in the quadratic formula calculator — it shows all three methods and checks every root.