The discriminant and the nature of roots

The number under the root

Look again at the formula:

Everything depends on the number under the square root. It discriminates between the kinds of roots, so it is called the discriminant:

Three cases

• $D > 0$. $\sqrt{D}$ is a positive number. The $+$ and the $-$ give two different answers: two distinct real roots.
• $D = 0$. $\sqrt{D} = 0$, and both signs give $x = -\dfrac{b}{2a}$: two equal real roots, the same number twice.
• $D < 0$. No real number has a negative square: no real roots.

On a graph, the roots are where the curve $y = ax^2 + bx + c$ meets the $x$-axis. Two crossings, one touch, or none: the three pictures match the three cases.

Using the discriminant

Example 1. What kind of roots does $2x^2 - 3x + 5 = 0$ have? $D = 9 - 40 = -31$. It is negative, so there are no real roots. We knew it without solving.

Example 2. For which values of $k$ does $x^2 + kx + 49 = 0$ have two equal roots?

1. Equal roots mean $D = 0$.
2. $D = k^2 - 4 \times 1 \times 49 = k^2 - 196$.
3. $k^2 = 196$, so $k = 14$ or $k = -14$.

Example 3. For which value of $k$ does $kx^2 + 12x + 4 = 0$ have two equal roots? $D = 144 - 16k = 0$, so $k = 9$. Then the equation is $9x^2 + 12x + 4 = 0$, which is $(3x + 2)^2 = 0$, with the root $x = -\dfrac{2}{3}$ twice.

Is the design possible?

A school wants a rectangular garden with a fence of $60\ \text{m}$ all round and an area of $250\ \text{m}^2$. Can it be done?

1. Half the perimeter is $30\ \text{m}$. If the length is $l$, the width is $30 - l$.
2. The area gives $l(30 - l) = 250$, that is $l^2 - 30l + 250 = 0$.
3. $D = 900 - 1000 = -100$. Negative, so no such garden exists.

With an area of $200\ \text{m}^2$ instead, $D = 900 - 800 = 100$. Then $l = \dfrac{30 \pm 10}{2}$: a garden of $20\ \text{m}$ by $10\ \text{m}$ is possible.

Practise with new numbers

Try it in Python

The function nature works out $D$ and names the kind of roots. The program tests four equations.

Common mistakes

• Saying "no roots" for $D = 0$. There is a root: the same number counted twice.
• Forgetting the second value of $k$. $k^2 = 196$ gives $k = 14$ and $k = -14$.
• Using $b^2 + 4ac$. The sign in the middle is a minus.
• Expecting two roots from every quadratic equation. When $D < 0$ there are no real roots at all.

Remember these

• $D = b^2 - 4ac$.
• $D > 0$: two distinct real roots. $D = 0$: two equal real roots. $D < 0$: no real roots.
• To make the roots equal, set $D = 0$ and solve for the unknown number.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Without solving, what kind of roots has $x^2 - 7x + 13 = 0$?

2. Find the positive $k$ for which $9x^2 + kx + 1 = 0$ has equal roots.

3. The graph of a quadratic expression just touches the $x$-axis. What is its discriminant?

4. Why does an equation with $a$ and $c$ of opposite signs always have two distinct real roots?

5. Can two numbers have a sum of $10$ and a product of $30$?