Graphical/numerical solutions | Year 9

01 / LEARN

The idea, step by step

A root is where the output is zero

To solve $x^{2} = 5$ graphically, plot $y = x^{2}$ and the horizontal line $y = 5$ . The $x$ coordinates of their intersections are the solutions. Equivalently, plot $y = x^{2} - 5$ and find its $x$ -intercepts.

A sketch gives estimates, so quote only the accuracy justified by its scale. A graph with one-unit grid squares does not justify six decimal places. A calculator or table can refine a graphical estimate.

Bracket and improve an estimate

For $f (x) = x^{2} - 5$ , $f (2.2) = - 0.16$ while $f (2.3) = 0.29$ . Because this polynomial graph is continuous, a root lies between $2.2$ and $2.3$ . Test a value inside the interval and keep the smaller interval containing a sign change.

Repeat until the required rounding is clear. Also check the other part of the graph: $x^{2} = 5$ has a negative solution as well as a positive one. A method must find all solutions in the stated domain.

A common trap

A graph estimate is automatically an exact answer.

Instead: A plotted intercept is approximate unless an exact calculation establishes it. State sensible rounding.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

Estimate both solutions of $x^{2} = 5$ to two decimal places.

The graph suggests roots near $- 2.2$ and $2.2$ .
${2.235}^{2} = 4.995225$ and ${2.237}^{2} = 5.004169$ , so the positive root lies between them.
Both bounds round to $2.24$ to two decimal places; symmetry gives $- 2.24$ .

x \approx - 2.24

2.24

Example 2

Worked example

Use substitution to locate the positive root of $x^{2} - 2 x - 1 = 0$ between two consecutive tenths.

At $2.4$ , $5.76 - 4.8 - 1 = - 0.04$ .
At $2.5$ , $6.25 - 5 - 1 = 0.25$ .
The continuous graph crosses zero between those inputs.

Between

2.4

and

2.5

Check your understanding

For $f (x) = x^{2} - 3$ , $f (1.7) = - 0.11$ and $f (1.8) = 0.24$ . Where is a positive root?

Show answer

Between $1.7$ and $1.8$ .

The continuous graph changes sign between those inputs.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Use a calculator to estimate both roots of $x^{2} = 7$ to one decimal place.
2 marks
For $f (x) = x^{2} - 6$ , calculate $f (2.4)$ and $f (2.5)$ .
2 marks
Explain what the previous outputs show about a root of $x^{2} - 6 = 0$ .
2 marks
The graph of $y = {(x - 1)}^{2} - 2$ is supplied. Estimate its roots to the nearest half unit.
2 marks
Use a table to locate the positive root of $x^{2} - 8 = 0$ between consecutive tenths.
3 marks
Why does finding $x \approx 2.45$ not fully solve $x^{2} = 6$ over the real numbers?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Use a calculator to estimate both roots of $x^{2} = 7$ to one decimal place.
2 marks
$\sqrt{7} \approx 2.64575$ , which rounds to $2.6$ .
$x \approx - 2.6, 2.6$
For $f (x) = x^{2} - 6$ , calculate $f (2.4)$ and $f (2.5)$ .
2 marks
${2.4}^{2} - 6 = 5.76 - 6 = - 0.24$ .
${2.5}^{2} - 6 = 6.25 - 6 = 0.25$ .
$- 0.24$ and $0.25$
Explain what the previous outputs show about a root of $x^{2} - 6 = 0$ .
2 marks
The continuous quadratic changes from negative to positive.
A positive root lies between $2.4$ and $2.5$ .
The graph of $y = {(x - 1)}^{2} - 2$ is supplied. Estimate its roots to the nearest half unit.
2 marks
The actual crossings are near $- 0.414$ and $2.414$ .
The nearest half-unit readings are $- 0.5$ and $2.5$ .
About $- 0.5$ and $2.5$ .
Use a table to locate the positive root of $x^{2} - 8 = 0$ between consecutive tenths.
3 marks
${2.8}^{2} - 8 = - 0.16$ and ${2.9}^{2} - 8 = 0.41$ .
Between $2.8$ and $2.9$ .
Why does finding $x \approx 2.45$ not fully solve $x^{2} = 6$ over the real numbers?
3 marks
Both a positive and its negative have the same square.
Check both sides of the parabola.
There is also $x \approx - 2.45$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

AC9M9A02, AC9M9A04–A06: monic quadratic expressions, integer-root equations, graphs and models.

Official curriculum source ↗

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