Sec 3 · G3 (Express)

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

Fractional equations: multiply by the LCM

\displaystyle x - \dfrac{2}{x} = 1

e.g.

\displaystyle x - \dfrac{2}{x} = 1

\displaystyle x^{2} - 2 = x

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

\displaystyle (x + 1)(x - 2) = 0

\displaystyle x = -1 \text{ or } x = 2

Check that no root makes a denominator $0$ .

📖 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.

x = 2 is one solution: substitute it to find k

\displaystyle 2x^{2} + kx - 6 = 0

e.g.

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

\displaystyle (x - 2)(2x + 3) = 0

$x = 2$ : $2(2)^{2} + 2k - 6 = 0$ , so $k = -1$ .
The other solution is $x = -\dfrac{3}{2}$ .

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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.)}

Coefficient of x² not 1

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

e.g.

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

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

\displaystyle = 2\left[\left(x + \dfrac{1}{2}\right)^{2} - \dfrac{1}{4}\right] - 3

\displaystyle = 2\left(x + \dfrac{1}{2}\right)^{2} - \dfrac{7}{2}

To solve $ax^{2} + bx + c = 0$ : divide every term by $a$ first.

A negative x² term: take out the minus first

\displaystyle -x^{2} + 6x - 2 = -(x^{2} - 6x) - 2

e.g.

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

\displaystyle = -\left((x - 3)^{2} - 9\right) - 2

\displaystyle = -(x - 3)^{2} + 7

📖 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

Story	Write in terms of $x$	Equation
number grid $w$ wide: a 2 by 2 square	smallest $x$ , largest $x + w + 1$	$x(x + w + 1) =$ product
working together (taps, painters)	in 1 hour: $\dfrac{1}{x}$ and $\dfrac{1}{x + d}$ of the job	$\dfrac{1}{x} + \dfrac{1}{x + d} = \dfrac{1}{T}$

📖 Worked example

The numbers $1$ , $2$ , $3$ , … are written in rows of $10$ on a grid. Part of the grid is shown. A square is drawn around four numbers, as shown, and it can be moved to other positions on the grid. In one position, the product of the smallest and the largest numbers in the square is $60$ . Let the smallest number be $x$ . Form an equation in $x$ and solve it to find the smallest number in the square.

1	$\boxed{2}$	$\boxed{3}$	4	5	6	7	8	9	10
11	$\boxed{12}$	$\boxed{13}$	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40

Step 1 of 8

Moving one row down adds the width of the grid. The largest number is

x + 11

📖 Worked example

Working on her own, Siti can paint a fence in $x$ hours. Her brother takes $6$ hours longer to paint the same fence on his own. Working together, they paint the fence in 1 hour 36 minutes. Form an equation in $x$ and solve it to find the time, in hours, her brother takes on his own.

Step 1 of 9

In one hour, the fraction of the job each does (1 ÷ the time it takes)

She:

\dfrac{1}{x}

Her brother:

\dfrac{1}{x + 6}

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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

7. Equation from the curve

Crosses the x-axis at a and b

\displaystyle y = (x - a)(x - b)

e.g.

\displaystyle y = (x - 1)(x - 4)

∩-shaped

\displaystyle y = -(x - a)(x - b)

Turning point (h, k)

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

e.g.

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

📖 Worked example

The diagram shows the curve $y = (x - a)(x - b)$ . Find the equation of the curve.

Step 1 of 4

Read the x-intercepts

The curve meets the

x

-axis at

x = -2

and

x = 3

📖 Worked example

The diagram shows the curve $y = -(x - h)^2 + k$ . Find the equation of the curve.

Step 1 of 4

Read the maximum point

Turning point:

x = -3

y = 5

📖 Worked example

The curve $y = ax^2 + bx + c$ passes through the points $(0, -7)$ and $(-1, 8)$ . Its line of symmetry is $x = -3$ . Find the values of $a$ , $b$ and $c$ , and write down the equation of the curve.

Step 1 of 6

The curve passes through (0, c): put x = 0

c = -7

Practise this →

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Sec 3 · G3 (Express)

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

Fractional equations: multiply by the LCM

\displaystyle x - \dfrac{2}{x} = 1

e.g.

\displaystyle x - \dfrac{2}{x} = 1

\displaystyle x^{2} - 2 = x

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

\displaystyle (x + 1)(x - 2) = 0

\displaystyle x = -1 \text{ or } x = 2

Check that no root makes a denominator $0$ .

📖 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.

x = 2 is one solution: substitute it to find k

\displaystyle 2x^{2} + kx - 6 = 0

e.g.

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

\displaystyle (x - 2)(2x + 3) = 0

$x = 2$ : $2(2)^{2} + 2k - 6 = 0$ , so $k = -1$ .
The other solution is $x = -\dfrac{3}{2}$ .

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.)}

Coefficient of x² not 1

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

e.g.

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

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

\displaystyle = 2\left[\left(x + \dfrac{1}{2}\right)^{2} - \dfrac{1}{4}\right] - 3

\displaystyle = 2\left(x + \dfrac{1}{2}\right)^{2} - \dfrac{7}{2}

To solve $ax^{2} + bx + c = 0$ : divide every term by $a$ first.

A negative x² term: take out the minus first

\displaystyle -x^{2} + 6x - 2 = -(x^{2} - 6x) - 2

e.g.

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

\displaystyle = -\left((x - 3)^{2} - 9\right) - 2

\displaystyle = -(x - 3)^{2} + 7

📖 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

Story	Write in terms of $x$	Equation
number grid $w$ wide: a 2 by 2 square	smallest $x$ , largest $x + w + 1$	$x(x + w + 1) =$ product
working together (taps, painters)	in 1 hour: $\dfrac{1}{x}$ and $\dfrac{1}{x + d}$ of the job	$\dfrac{1}{x} + \dfrac{1}{x + d} = \dfrac{1}{T}$

📖 Worked example

1	$\boxed{2}$	$\boxed{3}$	4	5	6	7	8	9	10
11	$\boxed{12}$	$\boxed{13}$	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40

Step 1 of 8

Moving one row down adds the width of the grid. The largest number is

x + 11

📖 Worked example

Step 1 of 9

In one hour, the fraction of the job each does (1 ÷ the time it takes)

She:

\dfrac{1}{x}

Her brother:

\dfrac{1}{x + 6}

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

7. Equation from the curve

Crosses the x-axis at a and b

\displaystyle y = (x - a)(x - b)

e.g.

\displaystyle y = (x - 1)(x - 4)

∩-shaped

\displaystyle y = -(x - a)(x - b)

Turning point (h, k)

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

e.g.

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

📖 Worked example

The diagram shows the curve $y = (x - a)(x - b)$ . Find the equation of the curve.

Step 1 of 4

Read the x-intercepts

The curve meets the

x

-axis at

x = -2

and

x = 3

📖 Worked example

The diagram shows the curve $y = -(x - h)^2 + k$ . Find the equation of the curve.

Step 1 of 4

Read the maximum point

Turning point:

x = -3

y = 5

📖 Worked example

The curve $y = ax^2 + bx + c$ passes through the points $(0, -7)$ and $(-1, 8)$ . Its line of symmetry is $x = -3$ . Find the values of $a$ , $b$ and $c$ , and write down the equation of the curve.

Step 1 of 6

The curve passes through (0, c): put x = 0

c = -7

Practise this →

Try a question: Quadratic equations and functions

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