How to solve quadratic equations: the quadratic formula and sum and product

A quadratic equation has the form $ax^2 + bx + c = 0$ with $a \neq 0$. There are three routes, and picking the right one saves time: factoring for incomplete equations, sum and product when the roots are integers, and the general formula for everything else.

Incomplete equations

• $b = 0$: $x^2 - 9 = 0 \Rightarrow x^2 = 9 \Rightarrow x = \pm 3$.
• $c = 0$: $x^2 - 4x = 0 \Rightarrow x\,(x - 4) = 0 \Rightarrow x = 0$ or $x = 4$.

Resist dividing $x^2 - 4x = 0$ by $x$: that erases the root $x = 0$.

The quadratic formula

The discriminant $\Delta$ tells you how many real roots there are:

Worked example: $x^2 - 5x + 6 = 0$

1. Coefficients: $a = 1$, $b = -5$, $c = 6$.
2. $\Delta = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1$.
3. $x = \dfrac{5 \pm 1}{2}$, so $x_1 = 3$ and $x_2 = 2$.

Worked example: $2x^2 + 3x - 2 = 0$

1. $a = 2$, $b = 3$, $c = -2$.
2. $\Delta = 9 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25$.
3. $x = \dfrac{-3 \pm 5}{4}$, so $x_1 = \dfrac{1}{2}$ and $x_2 = -2$.
4. Check: $2 \cdot \left(\tfrac{1}{2}\right)^2 + 3 \cdot \tfrac{1}{2} - 2 = \tfrac{1}{2} + \tfrac{3}{2} - 2 = 0$.

Worked example: $x^2 + 2x + 5 = 0$

1. $\Delta = 4 - 20 = -16 < 0$.
2. No real roots. Over the complex numbers, $x = \dfrac{-2 \pm 4i}{2} = -1 \pm 2i$.

Where the formula comes from

It comes from completing the square. Divide by $a$ and add $\left(\tfrac{b}{2a}\right)^2$ to both sides:

Take the square root of both sides and isolate $x$ to reach the formula. Knowing this helps you remember why the denominator is $2a$ and not just $2$.

Sum and product of the roots

For roots $x_1$ and $x_2$ of $ax^2 + bx + c = 0$:

Worked example: $x^2 - 7x + 12 = 0$

1. Sum $= 7$ and product $= 12$.
2. Pairs with product 12: $(1, 12)$, $(2, 6)$, $(3, 4)$. Only $(3, 4)$ adds to 7.
3. Roots: $3$ and $4$.

Worked example: $x^2 + x - 6 = 0$

1. Sum $= -1$ and product $= -6$. A negative product means the roots have opposite signs.
2. The pair $(2, -3)$ has sum $-1$ and product $-6$.
3. Roots: $2$ and $-3$.

Common mistakes

1. Forgetting that $b$ carries its sign: in $x^2 - 5x + 6$, $b = -5$ and $-b = 5$.
2. Computing $-5^2$ as $-25$ when the correct value is $(-5)^2 = 25$.
3. Dividing only $-b$ by $2a$ and forgetting the $\pm\sqrt{\Delta}$.