Mathematics

The quadratic formula

One formula solves every quadratic equation: the famous solution formula and what it reveals.

What you need first

Some quadratic equations yield to clever guessing, but with $2x^2 - 7x + 3 = 0$ the fun ends quickly. For exactly these cases there is a solution formula that always works: the quadratic formula. In German it is often called the midnight formula, because you should know it so well that you could recite it at midnight.

The shape of the equation

For the formula to apply, the equation must be in the form $a x^2 + b x + c = 0$ . You read off the three numbers $a$ , $b$ and $c$ , and the signs count. For $2x^2 - 7x + 3 = 0$ we have $a = 2$ , $b = -7$ and $c = 3$ . You put these three numbers into the solution formula, and it gives you the places where the meets the $x$ -axis.

x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The

\pm

stands for two calculations: once with plus, once with minus before the root.

The discriminant

The expression under the root has its own name: the discriminant $D = b^2 - 4ac$ . It decides the number of solutions before you compute any further. If $D > 0$ there are two solutions, the parabola crosses the $x$ -axis twice. If $D = 0$ both merge into one, the parabola only touches the axis. If $D < 0$ there is no real solution, because a negative number has no real square root.

D = b^2 - 4ac \quad\Rightarrow\quad D > 0:\ 2,\ \ D = 0:\ 1,\ \ D < 0:\ 0

A glance at

D

is enough: the number after the colon is how many real solutions there are.

Coordinate grid

y = 1x² + 1opens upwards

Try it: use the

c

slider to move the parabola up and down until it meets the

x

-axis twice, once or not at all. These are exactly the three cases the discriminant tells apart.

Step by step

Take $x^2 - 5x + 6 = 0$ with $a = 1$ , $b = -5$ , $c = 6$ . First the discriminant: $D = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1$ . Because $D > 0$ there are two solutions. The root of $1$ is $1$ , and since $-b = 5$ and $2a = 2$ , this reads $x = \frac{5 \pm 1}{2}$ . With plus you get $3$ , with minus you get $2$ . The solutions are $x_1 = 3$ and $x_2 = 2$ . And the equation from the start? For $2x^2 - 7x + 3 = 0$ we get $D = 49 - 24 = 25$ , the root is $5$ , so $x = \frac{7 \pm 5}{4}$ , which gives $x_1 = 3$ and $x_2 = 0.5$ .

Exercises

0 of 6 solved

Time to try it yourself. You can't break anything, every attempt counts.

What are $a$ , $b$ and $c$ in $x^2 - 5x + 6 = 0$ ?

Compute the discriminant $D = b^2 - 4ac$ for $x^2 - 5x + 6 = 0$ .

What is the larger solution of $x^2 - 5x + 6 = 0$ ?

Match each discriminant to the number of real solutions.

D > 0

D = 0

D < 0

How many real solutions does an equation with discriminant $D < 0$ have?

Put the steps for solving an equation with the quadratic formula in the correct order.

1Compute the discriminant D = b² − 4ac
2Substitute the values into the solution formula
3Read off a, b and c with their signs
4Bring the equation into the form a x² + b x + c = 0

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