Sec 2 · G3 (Express)

Quadratic equations and graphs

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

1. Solving by factorisation

Make one side 0, factorise, then each bracket = 0

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

e.g.

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

\displaystyle (x - 3)(x + 1) = 0

\displaystyle x = 3 \text{ or } x = -1

A common factor: never divide by x

\displaystyle x^{2} = 5x

e.g.

\displaystyle x^{2} - 5x = 0

\displaystyle x(x - 5) = 0

\displaystyle x = 0 \text{ or } x = 5

Coefficient of x² not 1

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

e.g.

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

\displaystyle (2x + 3)(x - 1) = 0

\displaystyle x = -\dfrac{3}{2} \text{ or } x = 1

📖 Worked example

Solve $x^{2} -5x = 6$ .

Step 1 of 3

Make one side zero

x^{2} - 5x - 6 = 0

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2. Graphs of quadratic functions

The curve y = x² − 2x − 3 crosses the x-axis at (−1, 0) and (3, 0), the y-axis at (0, −3), with minimum point (1, −4) on the line of symmetry x = 1.

Feature	How to find it	$y = x^{2} - 2x - 3$
shape	$\cup$ if the $x^{2}$ coefficient is positive, $\cap$ if negative	$\cup$
$y$ -intercept	put $x = 0$	$(0, -3)$
$x$ -intercepts	solve $y = 0$	$(-1, 0)$ , $(3, 0)$
line of symmetry	halfway between the $x$ -intercepts	$x = 1$
minimum point	on the line of symmetry	$(1, -4)$

📖 Worked example

Find the equation of the line of symmetry of the curve $y = (x + 2)(x + 3)$ .

Step 1 of 3

Which coordinate is 0 on the x-axis?

On the

x

-axis,

y = 0

(x + 2)(x + 3) = 0

Practise this →

3. Table of values for a quadratic

$x$	$-1$	$0$	$1$	$2$	$3$	$4$	$5$
$y = x^{2} - 4x + 3$	$8$	$3$	$0$	$-1$	$0$	$3$	$8$

Put a negative x in brackets, x = −1

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

e.g.

\displaystyle (-1)^{2} - 4(-1) + 3

\displaystyle = 1 + 4 + 3

\displaystyle = 8

The points from the table, (−1, 8), (0, 3), (1, 0), (2, −1), (3, 0), (4, 3) and (5, 8), joined by a smooth U-shaped curve with its minimum point (2, −1) on the line of symmetry x = 2.

📖 Worked example

Complete the table of values for $y = x^{2} - x + 2$ .

Step 1 of 2

Put each missing x into the equation and work out each term. Square a NEGATIVE x first: (−2)² = 4, not −4

When

x = -4

y = (-4)^2 - (-4) + 2

= 16 + 4 + 2

= 22

When

x = -1

y = (-1)^2 - (-1) + 2

= 1 + 1 + 2

= 4

When

x = 0

y = (0)^2 - (0) + 2

= 0 + 2

= 2

When

x = 3

y = (3)^2 - (3) + 2

= 9 - 3 + 2

= 8

📖 Worked example

Complete the table of values for $y = -x^{2} - 5$ .

Step 1 of 2

Put each missing x into the equation and work out each term. Square a NEGATIVE x first: (−2)² = 4, not −4; and −x² means −(x²)

When

x = -2

y = -(-2)^2 - 5

= -4 - 5

= -9

When

x = 1

y = -(1)^2 - 5

= -1 - 5

= -6

When

x = 4

y = -(4)^2 - 5

= -16 - 5

= -21

When

x = 5

y = -(5)^2 - 5

= -25 - 5

= -30

Practise this →

4. Reading a quadratic graph

The curve y = x² − 2x − 3. The line y = 5 meets it at x = −2 and x = 4; it crosses the x-axis at x = −1 and x = 3; its minimum point is (1, −4), on the line of symmetry x = 1.

Read	How	On $y = x^{2} - 2x - 3$
$y$ for a given $x$	up (or down) from $x$ to the curve, then across to the $y$ -axis	$x = -2$ : $y = 5$
$x$ for a given $y$	along the line $y = 5$ to the curve, then down to the $x$ -axis	$x = -2$ or $x = 4$
where it crosses the $x$ -axis	the points where $y = 0$	$x = -1$ or $x = 3$
minimum point	the lowest point	$(1, -4)$
line of symmetry	the vertical line through the minimum (or maximum) point	$x = 1$
maximum point ( $\cap$ curve)	the highest point	on $y = -x^{2} + 2x + 3$ : $(1, 4)$

📖 Worked example

The diagram shows the graph of $y = x^{2} + 2x + 1$ . Use the graph to find the values of $x$ when $y = 1$ .

Step 1 of 2

Draw the horizontal line y = 1 across the graph

The line

y = 1

meets the curve at two points.

📖 Worked example

The diagram shows the graph of $y = x^{2} - 2x - 3$ . Write down the coordinates of the minimum point of the curve.

Step 1 of 1

The minimum point is the lowest point of the curve: read across first, then up or down

The minimum point is

(1, -4)

Practise this →

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Sec 2 · G3 (Express)

Quadratic equations and graphs

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

Start free Worksheet (PDF)

1. Solving by factorisation

Make one side 0, factorise, then each bracket = 0

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

e.g.

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

\displaystyle (x - 3)(x + 1) = 0

\displaystyle x = 3 \text{ or } x = -1

A common factor: never divide by x

\displaystyle x^{2} = 5x

e.g.

\displaystyle x^{2} - 5x = 0

\displaystyle x(x - 5) = 0

\displaystyle x = 0 \text{ or } x = 5

Coefficient of x² not 1

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

e.g.

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

\displaystyle (2x + 3)(x - 1) = 0

\displaystyle x = -\dfrac{3}{2} \text{ or } x = 1

📖 Worked example

Solve $x^{2} -5x = 6$ .

Step 1 of 3

Make one side zero

x^{2} - 5x - 6 = 0

Practise this →

2. Graphs of quadratic functions

Feature	How to find it	$y = x^{2} - 2x - 3$
shape	$\cup$ if the $x^{2}$ coefficient is positive, $\cap$ if negative	$\cup$
$y$ -intercept	put $x = 0$	$(0, -3)$
$x$ -intercepts	solve $y = 0$	$(-1, 0)$ , $(3, 0)$
line of symmetry	halfway between the $x$ -intercepts	$x = 1$
minimum point	on the line of symmetry	$(1, -4)$

📖 Worked example

Find the equation of the line of symmetry of the curve $y = (x + 2)(x + 3)$ .

Step 1 of 3

Which coordinate is 0 on the x-axis?

On the

x

-axis,

y = 0

(x + 2)(x + 3) = 0

Practise this →

3. Table of values for a quadratic

$x$	$-1$	$0$	$1$	$2$	$3$	$4$	$5$
$y = x^{2} - 4x + 3$	$8$	$3$	$0$	$-1$	$0$	$3$	$8$

Put a negative x in brackets, x = −1

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

e.g.

\displaystyle (-1)^{2} - 4(-1) + 3

\displaystyle = 1 + 4 + 3

\displaystyle = 8

📖 Worked example

Complete the table of values for $y = x^{2} - x + 2$ .

Step 1 of 2

Put each missing x into the equation and work out each term. Square a NEGATIVE x first: (−2)² = 4, not −4

When

x = -4

y = (-4)^2 - (-4) + 2

= 16 + 4 + 2

= 22

When

x = -1

y = (-1)^2 - (-1) + 2

= 1 + 1 + 2

= 4

When

x = 0

y = (0)^2 - (0) + 2

= 0 + 2

= 2

When

x = 3

y = (3)^2 - (3) + 2

= 9 - 3 + 2

= 8

📖 Worked example

Complete the table of values for $y = -x^{2} - 5$ .

Step 1 of 2

Put each missing x into the equation and work out each term. Square a NEGATIVE x first: (−2)² = 4, not −4; and −x² means −(x²)

When

x = -2

y = -(-2)^2 - 5

= -4 - 5

= -9

When

x = 1

y = -(1)^2 - 5

= -1 - 5

= -6

When

x = 4

y = -(4)^2 - 5

= -16 - 5

= -21

When

x = 5

y = -(5)^2 - 5

= -25 - 5

= -30

Practise this →

4. Reading a quadratic graph

Read	How	On $y = x^{2} - 2x - 3$
$y$ for a given $x$	up (or down) from $x$ to the curve, then across to the $y$ -axis	$x = -2$ : $y = 5$
$x$ for a given $y$	along the line $y = 5$ to the curve, then down to the $x$ -axis	$x = -2$ or $x = 4$
where it crosses the $x$ -axis	the points where $y = 0$	$x = -1$ or $x = 3$
minimum point	the lowest point	$(1, -4)$
line of symmetry	the vertical line through the minimum (or maximum) point	$x = 1$
maximum point ( $\cap$ curve)	the highest point	on $y = -x^{2} + 2x + 3$ : $(1, 4)$

📖 Worked example

The diagram shows the graph of $y = x^{2} + 2x + 1$ . Use the graph to find the values of $x$ when $y = 1$ .

Step 1 of 2

Draw the horizontal line y = 1 across the graph

The line

y = 1

meets the curve at two points.

📖 Worked example

The diagram shows the graph of $y = x^{2} - 2x - 3$ . Write down the coordinates of the minimum point of the curve.

Step 1 of 1

The minimum point is the lowest point of the curve: read across first, then up or down

The minimum point is

(1, -4)

Practise this →

Try a question: Quadratic equations and graphs

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

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