Quadratic formula and completing the square

Visual model

Use the formula when factorising is not obvious

ax^{2} + bx + c = 0

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

identify a, b and c before substituting

Gold-standard guide

26 mins

What you will learn

Solve quadratics that do not factorise easily.

Use a clear step-by-step method for quadratic formula and completing the square.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier algebra skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a

Step 1

Identify a, b and c

a = 1, b = 6, c = 2

Step 2

Calculate the discriminant b^2 − 4ac

6² − 4(1)(2) = 36 − 8 = 28

Step 3

Substitute into the formula

x = (−6 ± $\sqrt{28}$ ) / 2

Watch out

Students forget to simplify the surd at the end, leaving $\sqrt{28}$ unresolved

Quadratic formula

$x = (-b \pm \sqrt{b^{2} - 4ac}) / 2a.$

Complete square

$x^{2} + bx = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2}.$

Worked example

Solve x² + 6x + 2 = 0, giving your answers in surd form.

Identify a, b and c: a = 1, b = 6, c = 2.

Calculate the discriminant b² − 4ac: 6² − 4(1)(2) = 36 − 8 = 28.

Substitute into the formula: x = (−6 ± $\sqrt{28}$ ) / 2.

Simplify the surd: $\sqrt{28}$ = $\sqrt{4 \times 7}$ = 2sqrt(7). So x = (−6 ± 2sqrt(7)) / 2 = −3 ± $\sqrt{7}$ .

Final answer

x = −3 + $\sqrt{7}$ or x = −3 − $\sqrt{7}$

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Solve x² + 6x + 2 = 0, giving your answers in surd form.

4 marks4 minsquadratic-formula-and-completing-the-square-worked

Show solution

Worked solution

1.Identify a, b and c: a = 1, b = 6, c = 2.
2.Calculate the discriminant b² − 4ac: 6² − 4(1)(2) = 36 − 8 = 28.
3.Substitute into the formula: x = (−6 ± $\sqrt{28}$ ) / 2.
4.Simplify the surd: $\sqrt{28}$ = $\sqrt{4 \times 7}$ = 2sqrt(7). So x = (−6 ± 2sqrt(7)) / 2 = −3 ± $\sqrt{7}$ .

Final answer

x = −3 + $\sqrt{7}$ or x = −3 − $\sqrt{7}$

Mark points

M1: identify a, b and c
M1: calculate the discriminant b² − 4ac
M1: substitute into the formula
M1: simplify the surd
A1: x = −3 + $\sqrt{7}$ or x = −3 − $\sqrt{7}$

Watch out

Students forget to simplify the surd at the end, leaving $\sqrt{28}$ unresolved.Divide both the integer part and the surd by any common factor.Also, the formula uses −b, so a positive b gives a negative numerator — watch the signs.

Diagnosticrecall

Use the formula to solve x² − 4x + 1 = 0. Leave in surd form.

1 mark2 minsquadratic-formula-and-completing-the-square-q1

Show solution

Worked solution

1.Spot the skill: Quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a.
2.Use the identify a, b and c stage first, then calculate the discriminant b² − 4ac.
3.Keep the final answer visible: x = 2 ± $\sqrt{3}$ .

Final answer

x = 2 ± $\sqrt{3}$

Mark points

M1: use the correct quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a. use it when the quadratic does not factorise.completing the square rewrites ax² + bx + c as a(x + p)² + q and directly gives the turning point (−p, q).
A1: x = 2 ± $\sqrt{3}$

Watch out

Easyprocedure

Complete the square for x² − 8x + 3, writing in the form (x + a)² + b.

2 marks3 minsquadratic-formula-and-completing-the-square-q2

Show solution

Worked solution

1.Spot the skill: Quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a.
2.Use the calculate the discriminant b² − 4ac stage first, then substitute into the formula.
3.Keep the final answer visible: (x − 4)² − 13.

Final answer

(x − 4)² − 13

Mark points

M1: use the correct quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a. use it when the quadratic does not factorise.completing the square rewrites ax² + bx + c as a(x + p)² + q and directly gives the turning point (−p, q).
A1: (x − 4)² − 13

Watch out

Mediumreasoning

Find the turning point of y = x² + 10x + 18 by completing the square.

3 marks4 minsquadratic-formula-and-completing-the-square-q3

Show solution

Worked solution

1.Spot the skill: Quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a.
2.Use the substitute into the formula stage first, then simplify the surd.
3.Keep the final answer visible: (−5, −7).

Final answer

(−5, −7)

Mark points

M1: use the correct quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a. use it when the quadratic does not factorise.completing the square rewrites ax² + bx + c as a(x + p)² + q and directly gives the turning point (−p, q).
A1: (−5, −7)

Watch out

Hardproblem solving

Show that x² + x + 1 = 0 has no real solutions.

3 marks5 minsquadratic-formula-and-completing-the-square-q4

Show solution

Worked solution

1.Spot the skill: Quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a.
2.Use the simplify the surd stage first, then identify a, b and c.
3.Keep the final answer visible: Discriminant = 1 − 4 = −3 < 0, no real solutions.

Final answer

Discriminant = 1 − 4 = −3 < 0, no real solutions

Mark points

M1: use the correct quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a. use it when the quadratic does not factorise.completing the square rewrites ax² + bx + c as a(x + p)² + q and directly gives the turning point (−p, q).
A1: Discriminant = 1 − 4 = −3 < 0, no real solutions

Watch out

Exam-stylemulti-step

Solve 3x² − 5x − 2 = 0 using the formula.

4 marks6 minsquadratic-formula-and-completing-the-square-q5

Show solution

Worked solution

1.Spot the skill: Quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a.
2.Use the identify a, b and c stage first, then calculate the discriminant b² − 4ac.
3.Keep the final answer visible: x = 2 or x = − $\frac{1}{3}$ .

Final answer

x = 2 or x = − $\frac{1}{3}$

Mark points

M1: use the correct quadratic formula: x = (−b ± $\sqrt{b^{2} − 4ac}$ ) / 2a. use it when the quadratic does not factorise.completing the square rewrites ax² + bx + c as a(x + p)² + q and directly gives the turning point (−p, q).
A1: x = 2 or x = − $\frac{1}{3}$

Watch out

Grade 9 stretchproblem solving

The graph y = x² - 6x + 1 crosses the x-axis at A and B. Find the exact x-coordinates of A and B, then find the minimum point of the graph.

4 marks7 minsquad-formula-g9

Show solution

Worked solution

1.Use the quadratic formula to solve x² - 6x + 1 = 0 exactly.
2.Complete the square to locate the turning point.
3.State both roots and the minimum as a coordinate.

Final answer

A and B have x-coordinates 3 - 2sqrt(2) and 3 + 2sqrt(2); the minimum point is (3, -8)

Mark points

M1: substitute a = 1, b = -6, c = 1 into the formula
A1: simplify the roots to 3 ± 2sqrt(2)
M1: write y = (x - 3)² - 8
A1: state the minimum point (3, -8)

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Quadratic formula and completing the square - 2 marksUse the formula to solve x² − 4x + 1 = 0. Leave in surd form.Mark answer

Answer

x = 2 ± $\sqrt{3}$

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer

Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer

Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer

Answer

3x(2x + 3)

Mastery check

I can explain the method for quadratic formula and completing the square.
I can show clear working without skipping key steps.
I can avoid this mistake: Students forget to simplify the surd at the end, leaving $\sqrt{28}$ unresolved.Divide both the integer part and the surd by any common factor.Also, the formula uses −b, so a positive b gives a negative numerator — watch the signs.

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Use the formula when factorising is not obvious

What you will learn

Know the rule, then use it

Method

Identify a, b and c

Calculate the discriminant b^2 − 4ac

Substitute into the formula

Watch out

Solve x2 + 6x + 2 = 0, giving your answers in surd form.

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.

Solve x² + 6x + 2 = 0, giving your answers in surd form.