Quadratic formula and exact surd answers | Year 11

01 / LEARN

The idea, step by step

A method for every quadratic

The formula follows from completing the square and applies whenever a is nonzero. First put the equation into $a x^{2} + b x + c = 0$ form.

Write coefficients with their signs. If b=-5, then -b=5 and b squared is 25. Keep the entire numerator over the denominator 2a.

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Exact before approximate

A surd is an exact value. Simplify square factors inside the root and then divide any factor common to the whole numerator and denominator. Never cancel with only one term of a sum.

For measurements, retain full precision until the final answer. State the rounding and reject a root only if the context rules it out; a negative algebraic root is not automatically wrong.

A common trap

Using denominator 2 and multiplying by a afterwards.

Instead: The single denominator is 2a; keep it under the whole numerator.

02 / WORKED EXAMPLES

See the thinking

Example 1

Irrational roots

Solve $x^{2} - 4 x - 1 = 0$ exactly.

Here a=1,b=-4,c=-1, giving discriminant 20.
$x = (4 \pm \sqrt{20}) / 2$ .
Use $\sqrt{20} = 2 \sqrt{5}$ and divide the whole numerator by 2.

x = 2 \pm \sqrt{5}

Example 2

A non-unit denominator

Solve $3 x^{2} + 2 x - 2 = 0$ exactly.

The discriminant is $4 + 24 = 28$ .
$x = (- 2 \pm \sqrt{28}) / 6 = (- 2 \pm 2 \sqrt{7}) / 6$ .
Reduce all terms by 2.

x = (- 1 \pm \sqrt{7}) / 3

Check your understanding

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

Show answer

$x = - 1 \pm \sqrt{3}$

The discriminant is 12, so the roots simplify to $(- 2 \pm 2 \sqrt{3}) / 2$ .

03 / WORKSHEET

Your turn

6 questions · 17 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Solve $x^{2} - 6 x + 1 = 0$ exactly.
3 marks
Solve $2 x^{2} + 4 x - 3 = 0$ exactly.
3 marks
Solve $3 x^{2} - 5 x + 1 = 0$ exactly.
3 marks
Solve $x^{2} = 3 x + 2$ exactly.
3 marks
A positive length x satisfies $x^{2} + 2 x = 8$ . Find x.
2 marks
Give both roots of $x^{2} - 2 x - 4 = 0$ to 3 decimal places.
3 marks

Read the worked solutions

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

Solve $x^{2} - 6 x + 1 = 0$ exactly.
3 marks
The discriminant is 32.
$x = (6 \pm 4 \sqrt{2}) / 2$ .
$x = 3 \pm 2 \sqrt{2}$
Solve $2 x^{2} + 4 x - 3 = 0$ exactly.
3 marks
The discriminant is 40.
$x = (- 4 \pm 2 \sqrt{10}) / 4$ .
$x = (- 2 \pm \sqrt{10}) / 2$
Solve $3 x^{2} - 5 x + 1 = 0$ exactly.
3 marks
Use a=3,b=-5,c=1.
The discriminant is 13.
$x = (5 \pm \sqrt{13}) / 6$
Solve $x^{2} = 3 x + 2$ exactly.
3 marks
Rearrange $x^{2} - 3 x - 2 = 0$ .
Use the quadratic formula.
$x = (3 \pm \sqrt{17}) / 2$
A positive length x satisfies $x^{2} + 2 x = 8$ . Find x.
2 marks
The equation is $x^{2} + 2 x - 8 = 0$ .
Roots are 2 and -4; only 2 is positive.
$x = 2$
Give both roots of $x^{2} - 2 x - 4 = 0$ to 3 decimal places.
3 marks
Exact roots are $1 \pm \sqrt{5}$ .
Evaluate both branches and round the final values.
$x \approx - 1.236, 3.236$

Checking saved progress…

Continue learning

Discriminant and the nature of roots

Next lesson

Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topics 2.1–2.2: quadratic relationships, cubic and quartic polynomials. Original Ascent teaching sequence.

Official curriculum source ↗

Back to Quadratics and polynomial graphs