SPI Mathematics · Quadratic Equations

Recognising a Quadratic Equation

A quadratic equation is one that can be written as ax² + bx + c = 0 with a not zero, so its highest power of x is 2.

Why this matters

Seven questions in the bank, three EASY and four MODERATE: name the degree, decide which equation is quadratic, or write a word situation as an equation. The trap in every one is deciding from the look of the equation before expanding it, because x² terms on the two sides often cancel.

Concept 1 of 3: Standard form of a quadratic equation

Every quadratic equation can be tidied into one shape: an x² term, an x term and a number, all on one side, with 0 on the other. Once it is in that shape you can read off a, b and c and see that the degree is 2.

Definition

Standard form: $ax^2 + bx + c = 0$ , where a, b, c are real numbers and $a \ne 0$ .
The degree is the highest power of x, so a quadratic has degree 2.
b or c may be 0: $x^2 - 9 = 0$ and $x^2 + 4x = 0$ are both quadratic.
If a were 0 there would be no $x^2$ term and the equation would be linear.
To reach standard form, move every term to one side. A term changes sign when it crosses the = sign.

Standard form

ax^2 + bx + c = 0, \quad a \ne 0

Worked example

Write

3x^2 - 7 = 2x

in standard form and give a, b and c.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Quadratic Equations · Recognising a Quadratic Equation

5x^2 + 8x + 4 = 2x^2 + 4x + 6

is a

Thinking b or c must be non-zero

Only a must be non-zero.

x^2 - 16 = 0

has

b = 0

and is still a quadratic equation of degree 2.

Keeping the sign when a term crosses the = sign

A term changes sign when it moves to the other side:

x^2 = 3x + 4

becomes

x^2 - 3x - 4 = 0

, not

x^2 + 3x + 4 = 0

Drill 2 more on standard form of a quadratic equation

Concept 2 of 3: Simplifying both sides before naming the type of equation

An equation can look quadratic and turn out linear, because the

x^2

on one side cancels the

x^2

on the other. It can also look like a cubic and turn out quadratic. Expand first, collect on one side, and only then look at the highest power left.

Definition

Expand every bracket, move all terms to one side and collect like terms.
If the $x^2$ terms cancel and only x is left, the equation is linear.
If an $x^3$ term survives, it is a cubic. If the $x^3$ terms cancel and $x^2$ is left, it is quadratic.
A term in $\sqrt{x}$ is not a polynomial term, so such an equation is not quadratic. A number under the root is fine: $\sqrt{2}x^2$ is allowed.
A term in $\frac{1}{x}$ : multiply every term by the power of x that clears it, then check the degree.

Worked example

(x + 3)^2 = x^2 + 4x + 1

a quadratic equation?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Quadratic Equations · Recognising a Quadratic Equation

Which of the given equation is not quadratic?

Judging by the look before expanding

x^2

on both sides proves nothing.

x(x + 3) = x^2 - 5

loses its

x^2

terms and reduces to

3x + 5 = 0

, a linear equation.

Confusing a root in the coefficient with a root of x

\sqrt{3}x^2 + x - 1 = 0

is quadratic, because

\sqrt{3}

is just a number.

x^2 + \sqrt{x} = 1

is not, because x itself is under the root.

Drill 2 more on simplifying both sides before naming the type of equation

Concept 3 of 3: Writing a word situation as a quadratic equation

A word problem becomes an equation in three moves: call the unknown x, write every other quantity using x, then use the one fact the problem gives (an area, a product, a right angle) to join them with an = sign.

Definition

Let the smallest or simplest unknown be x, and write the others from it: '6 more than twice x' is $2x + 6$ .
Rectangle: length $\times$ breadth $=$ area.
Consecutive integers: x and $x + 1$ .
Right triangle: the hypotenuse is the longest side, and its square equals the sum of the squares of the other two sides.

Pythagoras theorem

(\text{hypotenuse})^2 = (\text{base})^2 + (\text{height})^2

Worked example

The length of a rectangle is 3 m more than its breadth, and its area is 40 m². Write a quadratic equation for the breadth x.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Quadratic Equations · Recognising a Quadratic Equation

The hypotenuse of a right triangle is 6m more than twice the shortest side. The third side is 2m less than the hypotenuse. The representation of the above situation in the form of a quadratic equation is

Adding the hypotenuse to another side

The hypotenuse stands alone on one side:

\text{hypotenuse}^2 = a^2 + b^2

. An equation that adds the hypotenuse squared to another side squared is wrong.

Reading 'more than' backwards

'6 more than twice x' is

2x + 6

, and '2 less than h' is

h - 2

. Write the base quantity first, then add or take away.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Standard form of a quadratic equation
Standard form
$ax^2 + bx + c = 0, \quad a \ne 0$
Writing a word situation as a quadratic equation
Pythagoras theorem
$(\text{hypotenuse})^2 = (\text{base})^2 + (\text{height})^2$

Watch out for (6)

Thinking b or c must be non-zero
→ Standard form of a quadratic equation
Keeping the sign when a term crosses the = sign
→ Standard form of a quadratic equation
Judging by the look before expanding
→ Simplifying both sides before naming the type of equation
Confusing a root in the coefficient with a root of x
→ Simplifying both sides before naming the type of equation
Adding the hypotenuse to another side
→ Writing a word situation as a quadratic equation
Reading 'more than' backwards
→ Writing a word situation as a quadratic equation

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