What makes an equation quadratic

A new kind of equation

Nisha has $60\ \text{m}$ of fencing. She wants a rectangular vegetable bed against a wall, so only three sides need fencing. If each short side is $x$ metres, the long side is $60 - 2x$ metres, and the area is $x(60 - 2x)$ square metres.

For an area of $400\ \text{m}^2$, she needs

Multiply out: $60x - 2x^2 = 400$. The equation has an $x^2$ in it. It is not a linear equation like those of Class 9. It is a quadratic equation.

The standard form

A quadratic equation in $x$ is an equation that can be written as

where $a$, $b$ and $c$ are real numbers and $a \neq 0$. This is the standard form.

• If $a$ were $0$, the $x^2$ would vanish, and the equation would be linear.
• $b$ or $c$ can be $0$. So $x^2 - 9 = 0$ and $3x^2 + 4x = 0$ are quadratic equations too.
• The left side, $ax^2 + bx + c$, is a quadratic polynomial: a polynomial of degree $2$.

Nisha's equation becomes $-2x^2 + 60x - 400 = 0$. Dividing every term by $-2$ gives the neater $x^2 - 30x + 200 = 0$.

Expand before you decide

An equation can look quadratic and not be one. Always multiply out the brackets and collect the terms on one side first.

A worked example. Is $(x + 4)^2 = x^2 + 2x$ a quadratic equation?

1. Expand. $(x + 4)^2 = x^2 + 8x + 16$.
2. Collect. $x^2 + 8x + 16 - x^2 - 2x = 0$, so $6x + 16 = 0$.
3. Decide. The $x^2$ terms cancel. The equation is linear, not quadratic.

Now try $x(2x + 1) = x^2 - 3$. It becomes $2x^2 + x = x^2 - 3$, that is $x^2 + x + 3 = 0$. The $x^2$ term stays, so it is quadratic.

Roots of a quadratic equation

A number is a root of $ax^2 + bx + c = 0$ when putting it in place of $x$ makes the left side equal to $0$. The roots of the equation are also called the zeroes of the polynomial $ax^2 + bx + c$.

Test a root by putting it in. Is $x = 4$ a root of $x^2 - 7x + 12 = 0$?

Yes, it is. And $x = 5$? $25 - 35 + 12 = 2$, which is not $0$, so $5$ is not a root.

A quadratic equation has at most two roots. On a graph, the roots are the points where the curve $y = ax^2 + bx + c$ meets the $x$ axis.

Practise with new numbers

Try it in Python

This program tests a list of numbers against the equation $2x^2 - 7x + 3 = 0$ and says which of them are roots.

Common mistakes

• Deciding before simplifying. $(x + 4)^2 = x^2 + 2x$ looks quadratic, but the $x^2$ terms cancel.
• Forgetting that $a$ must not be $0$. $0x^2 + 3x - 2 = 0$ is a linear equation.
• Changing a sign wrongly. When a term moves to the other side, its sign changes: $x^2 = 3x - 5$ becomes $x^2 - 3x + 5 = 0$.
• Testing a root carelessly. Square first, then multiply, then add: for $x = -3$, $x^2$ is $+9$, not $-9$.

Remember these

• A quadratic equation can be written as $ax^2 + bx + c = 0$ with $a \neq 0$.
• Expand and simplify before you decide whether an equation is quadratic.
• A root makes the left side $0$. Put it in to check.
• A quadratic equation has at most two roots: where its curve meets the $x$ axis.

Check yourself

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

1. Is $(x + 3)^2 = x^2 - 6$ a quadratic equation?

2. Write $2x(x - 1) = 3x + 4$ in standard form, and give $a$, $b$ and $c$.

3. Is $x = -2$ a root of $x^2 + 5x + 6 = 0$?

4. A plot is $6\ \text{m}$ longer than it is wide, and its area is $112\ \text{m}^2$. Write a quadratic equation for its width $w$.

5. Why does the definition of a quadratic equation say $a \neq 0$?