Mathematics

Quadratic equations

Solving equations with x squared: taking roots, finding zeros and factoring step by step.

What you need first

The flight path of a thrown ball, the area of a square bed, the braking distance of a car: as soon as a quantity is multiplied by itself, an $x^2$ shows up. Quadratic equations show you how to work such tasks backwards and find the value you are looking for.

What is a quadratic equation?

A quadratic equation is an equation in which the unknown appears squared, that is, as $x^2$ . The simplest example is $x^2 = 9$ . We are looking for the number that gives 9 when multiplied by itself. That is 3, because $3 \cdot 3 = 9$ . But −3 works too, because minus times minus gives plus. Quadratic equations therefore often have two solutions.

Solving by taking the root

If the equation has the form $x^2 = a$ , you take the square root on both sides. For $x^2 = 25$ we get $x = 5$ or $x = -5$ . Important: do not forget the negative solution. If something is added on top, move it to the other side first. $x^2 + 4 = 20$ becomes $x^2 = 16$ , so $x = 4$ or $x = -4$ . If a number stands in front of the $x^2$ as a factor, divide by it first: $3x^2 = 27$ becomes $x^2 = 9$ , so $x = 3$ or $x = -3$ . If the right side is a negative number instead, as in $x^2 = -4$ , there is no solution at all, because a square is never negative.

x^2 = a \quad\Rightarrow\quad x = \pm\sqrt{a}

The basic rule: the square root undoes the square, with both plus and minus.

Coordinate grid

y = 1x² + 1opens upwards

Here the two sliders are called

a

and

c

: they draw the parabola

y = a x^2 + c

. Push

c

down until the curve meets the x-axis. That is exactly where the zeros sit, the solutions of

a x^2 + c = 0

Zeros: where the graph meets the x-axis

Every quadratic equation can also be read as a picture. To do that, first bring everything to one side, so that only a 0 is left on the right. $x^2 = 9$ then becomes $x^2 - 9 = 0$ , and the left side belongs to the $y = x^2 - 9$ . The solutions of the equation are exactly the places where this meets the x-axis, here at $x = 3$ and $x = -3$ . They are called zeros because $y = 0$ there. A can cross the axis twice, just touch it, or miss it completely. In the same way the equation has two, one, or no solution.

Solving by factoring

If the right side is 0 and every term on the left contains an x, factoring helps. In $x^2 - 5x = 0$ both parts contain an x. You write $x \cdot (x - 5) = 0$ . A product is zero exactly when one of the factors is zero. So $x = 0$ or $x - 5 = 0$ , which gives $x = 0$ or $x = 5$ .

Exercises

0 of 6 solved

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

Which number is a solution of x² = 16?

Solve x² = 36. Give the positive solution.

Solve x² = 81. Give the positive solution.

When taking the root of x² = 25, you must not forget the … solution.

The equation x · (x − 5) = 0 has the solutions 0 and which number?

Put the steps for solving x² + 4 = 20 in the correct order.

1Subtract 4 on both sides: x² = 16
2Take the root: x = 4 or x = −4
3Write the equation: x² + 4 = 20

Where this leads

Polynomial division

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