Sec 3 · G2 (N(A))

Quadratic equations and functions

Practise quadratic equations and functions with step-by-step help. Free account.

1. Solving by factorisation

Zero product

\displaystyle AB = 0 \;\Rightarrow\; A = 0 \text{ or } B = 0

e.g.

\displaystyle x^{2} - 8x + 12 = 0

\displaystyle (x - 2)(x - 6) = 0

\displaystyle x - 2 = 0 \text{ or } x - 6 = 0

\displaystyle x = 2 \text{ or } x = 6

📖 Worked example

Solve $x^{2} - 8x + 12 = 0$ .

Step 1 of 2

Factorise

(x - 6)(x - 2) = 0

📖 Worked example

Solve $x^{2} = 7x$ .

Step 1 of 3

Make one side zero

x^{2} - 7x = 0

Dividing both sides by

x

would lose a solution, so don't.

Practise this →

2. Completing the square

Completing the square

\displaystyle x^{2} + bx = \left(x + \dfrac{b}{2}\right)^{2} - \left(\dfrac{b}{2}\right)^{2}

e.g.

\displaystyle x^{2} + 6x - 5

\displaystyle = (x + 3)^{2} - 3^{2} - 5

\displaystyle = (x + 3)^{2} - 14

Solving

\displaystyle (x + h)^{2} = k \;\Rightarrow\; x + h = \pm\sqrt{k}

e.g.

\displaystyle x^{2} - 6x - 4 = 0

\displaystyle (x - 3)^{2} - 13 = 0

\displaystyle (x - 3)^{2} = 13

\displaystyle x - 3 = \pm\sqrt{13}

\displaystyle x = 3 \pm \sqrt{13}

\displaystyle x = 6.61 \text{ or } x = -0.61\text{ (2 d.p.)}

📖 Worked example

Express $x^{2} + 6x - 5$ in the form $(x + h)^2 + k$ .

Step 1 of 2

Halve the coefficient of x

Half of

6

3

, so the bracket is

(x + 3)

📖 Worked example

Solve $x^{2} - 10x - 1 = 0$ by completing the square, giving your answers correct to 2 decimal places.

Step 1 of 4

Halve the coefficient of x

Half of

-10

-5

, so the bracket is

(x - 5)

Practise this →

3. The quadratic formula

Quadratic formula, for ax² + bx + c = 0

\displaystyle x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

e.g.

\displaystyle 2x^{2} + 3x - 4 = 0

\displaystyle x = \dfrac{-3 \pm \sqrt{3^{2} - 4(2)(-4)}}{2(2)}

\displaystyle x = \dfrac{-3 \pm \sqrt{41}}{4}

\displaystyle x = 0.85 \text{ or } x = -2.35\text{ (2 d.p.)}

📖 Worked example

Solve $x^{2} + 6x + 2 = 0$ , giving your answers correct to 2 decimal places.

Step 1 of 3

Read off a, b and c

a = 1,\ b = 6,\ c = 2

📖 Worked example

Solve $2x^{2} - 7x + 4 = 0$ , giving your answers correct to 2 decimal places.

Step 1 of 3

Read off a, b and c

a = 2,\ b = -7,\ c = 4

Practise this →

4. Word problems

Five steps

\displaystyle \text{let } x \;\to\; \text{form} \;\to\; \text{solve} \;\to\; \text{reject} \;\to\; \text{answer}

e.g.

\displaystyle x(x + 4) = 96

\displaystyle x^{2} + 4x - 96 = 0

\displaystyle (x - 8)(x + 12) = 0

\displaystyle x = 8 \text{ or } x = -12

Let $x$ be one unknown, with its unit: width $= x$ cm.
Form the equation from the story, then make one side $0$ .
Solve: factorise, or use the formula.
Reject the root that can't be: a negative length, a side that would be negative.
Answer the question asked: the length $x + 4$ , not $x$ .

📖 Worked example

The length of a rectangle is $5$ cm more than its width. Its area is $104$ cm². Let the width be $x$ cm. Find the length of the rectangle.

Step 1 of 7

Which relationship gives the equation?

length × width = area

Practise this →

5. Graphs of quadratic functions

Graph of y = x squared minus 4x plus 3: crosses the y-axis at (0, 3), the x-axis at 1 and 3, line of symmetry x = 2, minimum point (2, −1).

Feature	How	$y = x^{2} - 4x + 3$
$y$ -intercept	$x = 0$	$(0, 3)$
$x$ -intercepts	$y = 0$	$x = 1,\ x = 3$
Line of symmetry	$x = \dfrac{x_{1} + x_{2}}{2}$	$x = 2$
Turning point	$(x - h)^{2} + k \;\to\; (h, k)$	$(2, -1)$

Left: U-shaped curve y = (x − 1)² + 2 with a minimum point (1, 2). Right: ∩-shaped curve y = −(x − 1)² + 3 with a maximum point (1, 3).

📖 Worked example

Find the $x$ -coordinates of the points where the curve $y = x^{2} - 4x + 3$ meets the $x$ -axis.

Step 1 of 3

Which coordinate is 0 on the x-axis?

On the

x

-axis,

y = 0

x^{2} - 4x + 3 = 0

📖 Worked example

Find the coordinates of the turning point of the curve $y = x^{2} - 4x - 5$ .

Step 1 of 2

Complete the square

x^{2} - 4x - 5 = (x - 2)^2 - 4 - 5

= (x - 2)^2 - 9

Practise this →

6. The three forms of a quadratic

General form

\displaystyle y = ax^{2} + bx + c

y = x squared minus 2x minus 3: U-shaped, crosses the y-axis at (0, −3).

$a > 0$ : ∪-shaped, $a < 0$ : ∩-shaped
$y$ -intercept: $(0, c)$

Factorised form

\displaystyle y = \pm(x - p)(x - q)

y = (x + 1)(x − 3): the same curve, crossing the x-axis at (−1, 0) and (3, 0).

$x$ -intercepts: $(p, 0)$ and $(q, 0)$
Line of symmetry: $x = \dfrac{p + q}{2}$

Completed-square form

\displaystyle y = \pm(x - h)^{2} + k

y = (x − 1)² − 4: the same curve, minimum point (1, −4), line of symmetry x = 1.

Turning point: $(h, k)$
Line of symmetry: $x = h$
$+$ : minimum point, $-$ : maximum point

General → factorised: factorise

\displaystyle x^{2} - 2x - 3

e.g.

\displaystyle x^{2} - 2x - 3

\displaystyle = (x + 1)(x - 3)

General → completed square

\displaystyle x^{2} - 2x - 3

e.g.

\displaystyle x^{2} - 2x - 3

\displaystyle = (x - 1)^{2} - 1^{2} - 3

\displaystyle = (x - 1)^{2} - 4

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Sec 3 · G2 (N(A))

Quadratic equations and functions

Practise quadratic equations and functions with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Solving by factorisation

Zero product

\displaystyle AB = 0 \;\Rightarrow\; A = 0 \text{ or } B = 0

e.g.

\displaystyle x^{2} - 8x + 12 = 0

\displaystyle (x - 2)(x - 6) = 0

\displaystyle x - 2 = 0 \text{ or } x - 6 = 0

\displaystyle x = 2 \text{ or } x = 6

📖 Worked example

Solve $x^{2} - 8x + 12 = 0$ .

Step 1 of 2

Factorise

(x - 6)(x - 2) = 0

📖 Worked example

Solve $x^{2} = 7x$ .

Step 1 of 3

Make one side zero

x^{2} - 7x = 0

Dividing both sides by

x

would lose a solution, so don't.

Practise this →

2. Completing the square

Completing the square

\displaystyle x^{2} + bx = \left(x + \dfrac{b}{2}\right)^{2} - \left(\dfrac{b}{2}\right)^{2}

e.g.

\displaystyle x^{2} + 6x - 5

\displaystyle = (x + 3)^{2} - 3^{2} - 5

\displaystyle = (x + 3)^{2} - 14

Solving

\displaystyle (x + h)^{2} = k \;\Rightarrow\; x + h = \pm\sqrt{k}

e.g.

\displaystyle x^{2} - 6x - 4 = 0

\displaystyle (x - 3)^{2} - 13 = 0

\displaystyle (x - 3)^{2} = 13

\displaystyle x - 3 = \pm\sqrt{13}

\displaystyle x = 3 \pm \sqrt{13}

\displaystyle x = 6.61 \text{ or } x = -0.61\text{ (2 d.p.)}

📖 Worked example

Express $x^{2} + 6x - 5$ in the form $(x + h)^2 + k$ .

Step 1 of 2

Halve the coefficient of x

Half of

6

3

, so the bracket is

(x + 3)

📖 Worked example

Solve $x^{2} - 10x - 1 = 0$ by completing the square, giving your answers correct to 2 decimal places.

Step 1 of 4

Halve the coefficient of x

Half of

-10

-5

, so the bracket is

(x - 5)

Practise this →

3. The quadratic formula

Quadratic formula, for ax² + bx + c = 0

\displaystyle x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

e.g.

\displaystyle 2x^{2} + 3x - 4 = 0

\displaystyle x = \dfrac{-3 \pm \sqrt{3^{2} - 4(2)(-4)}}{2(2)}

\displaystyle x = \dfrac{-3 \pm \sqrt{41}}{4}

\displaystyle x = 0.85 \text{ or } x = -2.35\text{ (2 d.p.)}

📖 Worked example

Solve $x^{2} + 6x + 2 = 0$ , giving your answers correct to 2 decimal places.

Step 1 of 3

Read off a, b and c

a = 1,\ b = 6,\ c = 2

📖 Worked example

Solve $2x^{2} - 7x + 4 = 0$ , giving your answers correct to 2 decimal places.

Step 1 of 3

Read off a, b and c

a = 2,\ b = -7,\ c = 4

Practise this →

4. Word problems

Five steps

\displaystyle \text{let } x \;\to\; \text{form} \;\to\; \text{solve} \;\to\; \text{reject} \;\to\; \text{answer}

e.g.

\displaystyle x(x + 4) = 96

\displaystyle x^{2} + 4x - 96 = 0

\displaystyle (x - 8)(x + 12) = 0

\displaystyle x = 8 \text{ or } x = -12

Let $x$ be one unknown, with its unit: width $= x$ cm.
Form the equation from the story, then make one side $0$ .
Solve: factorise, or use the formula.
Reject the root that can't be: a negative length, a side that would be negative.
Answer the question asked: the length $x + 4$ , not $x$ .

📖 Worked example

The length of a rectangle is $5$ cm more than its width. Its area is $104$ cm². Let the width be $x$ cm. Find the length of the rectangle.

Step 1 of 7

Which relationship gives the equation?

length × width = area

Practise this →

5. Graphs of quadratic functions

Feature	How	$y = x^{2} - 4x + 3$
$y$ -intercept	$x = 0$	$(0, 3)$
$x$ -intercepts	$y = 0$	$x = 1,\ x = 3$
Line of symmetry	$x = \dfrac{x_{1} + x_{2}}{2}$	$x = 2$
Turning point	$(x - h)^{2} + k \;\to\; (h, k)$	$(2, -1)$

📖 Worked example

Find the $x$ -coordinates of the points where the curve $y = x^{2} - 4x + 3$ meets the $x$ -axis.

Step 1 of 3

Which coordinate is 0 on the x-axis?

On the

x

-axis,

y = 0

x^{2} - 4x + 3 = 0

📖 Worked example

Find the coordinates of the turning point of the curve $y = x^{2} - 4x - 5$ .

Step 1 of 2

Complete the square

x^{2} - 4x - 5 = (x - 2)^2 - 4 - 5

= (x - 2)^2 - 9

Practise this →

6. The three forms of a quadratic

General form

\displaystyle y = ax^{2} + bx + c

$a > 0$ : ∪-shaped, $a < 0$ : ∩-shaped
$y$ -intercept: $(0, c)$

Factorised form

\displaystyle y = \pm(x - p)(x - q)

$x$ -intercepts: $(p, 0)$ and $(q, 0)$
Line of symmetry: $x = \dfrac{p + q}{2}$

Completed-square form

\displaystyle y = \pm(x - h)^{2} + k

Turning point: $(h, k)$
Line of symmetry: $x = h$
$+$ : minimum point, $-$ : maximum point

General → factorised: factorise

\displaystyle x^{2} - 2x - 3

e.g.

\displaystyle x^{2} - 2x - 3

\displaystyle = (x + 1)(x - 3)

General → completed square

\displaystyle x^{2} - 2x - 3

e.g.

\displaystyle x^{2} - 2x - 3

\displaystyle = (x - 1)^{2} - 1^{2} - 3

\displaystyle = (x - 1)^{2} - 4

Try a question: Quadratic equations and functions

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise quadratic equations and functions with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free