Completing the square and the quadratic formula, solve any quadratic
Complete the square and use the quadratic formula. Worked examples, exam-style questions with answers.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
Download the PDF (652 KB)What do you do when a quadratic won't factorise?
Completing the square and the quadratic formula feel hard because quadratics don’t always factorise. Imagine a square with a corner missing. You fix it by adding and subtracting the same amount.
The picture to keep in your head: The square with a corner missing
Picture x2 + 6x as tiles: one x-by-x square and six x-by-1 strips. Split the strips into two groups of three and put one group along each side of the square. You almost have a bigger square, (x + 3) by (x + 3), but its corner is missing: a 3-by-3 gap of 9 tiles. So x2 + 6x = (x + 3)2 - 9: complete the square, then take away the corner you added. The quadratic formula is this same trick done once for every quadratic.
The method, every time
- Halve the x-term and square it. Add and subtract that amount to keep the equation balanced.
- Rewrite the equation as a square minus the added amount. This is completing the square.
- Take the square root of both sides. Remember to keep the ± symbol for two answers.
- Use the quadratic formula when factorising fails. It works for any quadratic.
- Check your answers by substituting back into the original equation.
Words you'll see
- complete the square
- Rewrite a quadratic by adding and subtracting the same amount to form a perfect square.
- quadratic formula
- A formula that solves any quadratic equation using a, b, and c.
- factorise
- Break a quadratic into two brackets that multiply to give the original expression.
- square root
- Find a number that multiplies by itself to give the original number.
- discriminant
- b2 - 4ac: positive means two solutions, zero one, negative none
- discriminant
- b2 - 4ac: positive means two solutions, zero one, negative none
Watch it explained
- The 10-second trick for factorising quadratics (one-minute video)
Halve, square, take away
Imagine x² + 6x as a square with a corner missing. To complete the square, you halve the x-term (6 ÷ 2 = 3) and make a perfect square (x + 3)². But this adds 9 tiles, so you must take 9 away. For x² + bx + c, halve b, square it, then subtract it to keep the equation balanced.
Watch out
- Forgetting to subtract the square of half b. Always subtract (b/2)² after making the square
- Halving the wrong number. Only halve the coefficient of x, not the constant term
Square root both sides, keep the ±
Imagine x² + 4x - 1 as a square with a corner missing. To solve it, complete the square, move the number across, then square root both sides. Remember to keep the ± sign. The turning point of the graph is (-p, q).
Watch out
- Losing the negative root. Always include ± when taking the square root.
- Rounding too early. Round only at the end to keep accuracy.
One formula for every quadratic
The quadratic formula is the same trick as completing the square, but done once for every quadratic. Imagine you have a square with a corner missing — the formula fills in that corner and gives you the answers. The formula is x equals negative b plus or minus the square root of b squared minus four a c, all over two a. This works for any quadratic equation.
Watch out
- Squaring a negative b without brackets, e.g. -32 = -9. Use brackets: (-3)2 = 9
- Dividing only the square root by 2a. Divide the whole numerator by 2a
Factorise, complete the square, or formula
Imagine a square with a corner missing. Completing the square is like filling in the missing corner, and the quadratic formula is the same trick done automatically. Factorise when numbers work nicely, use the formula when you need decimals or surds, and complete the square when you need a vertex form. The discriminant, b² − 4ac, tells you how many solutions there are: positive for two, zero for one, and negative for none.
Watch out
- Trying to factorise a quadratic that won't factorise nicely. Use the quadratic formula or complete the square instead.
- Giving 'no solution' when b² − 4ac is zero. One real solution when discriminant is zero.
20 exam-style completing the square and the quadratic formula questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Complete the square (Higher)
- 1Write x² + 6x + 5 in the form (x + p)² + q.[2 marks]
Show the answer
(x + 3)2 - 4
x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4 - 2Write x² + 4x + 3 in the form (x + p)² + q.[2 marks]
Show the answer
(x + 2)2 - 1
x² + 4x + 3 = (x + 2)² - 4 + 3 = (x + 2)² - 1 - 3Write x² + 5x + 6 in the form (x + p)² + q.[2 marks]
Show the answer
(x + 2.5)2 - 0.25
x² + 5x + 6 = (x + 2.5)² - 6.25 + 6 = (x + 2.5)² - 0.25 - 4Write x² - 4x + 3 in the form (x + p)² + q.[2 marks]
Show the answer
(x - 2)2 - 1
x² - 4x + 3 = (x - 2)² - 4 + 3 = (x - 2)² - 1
B · Turning points (Higher)
- 5Find the turning point of y = x² - 8x + 3.[3 marks]
Show the answer
(4, -13)
Complete the square: x² - 8x = (x - 4)² - 16. So y = (x - 4)² - 13. Turning point is (4, -13). - 6Find the minimum value of y = x² + 6x + 5.[3 marks]
Show the answer
-4
Complete the square: x² + 6x = (x + 3)² - 9. So y = (x + 3)² - 4. Minimum value is -4. - 7Find the turning point of y = x² + 2x - 15.[3 marks]
Show the answer
(-1, -16)
Complete the square: x² + 2x = (x + 1)² - 1. So y = (x + 1)² - 16. Turning point is (-1, -16).
C · Solve by completing the square (Higher)
- 8Solve x² + 6x + 5 = 0 by completing the square.[3 marks]
Show the answer
x = -1 or x = -5
x² + 6x + 5 = 0. Complete the square: x² + 6x + 9 = 4. (x + 3)² = 4. x + 3 = ±2. x = -3 ± 2. - 9Solve x² + 4x + 3 = 0 by completing the square.[3 marks]
Show the answer
x = -1 or x = -3
x² + 4x + 3 = 0. Complete the square: x² + 4x + 4 = 1. (x + 2)² = 1. x + 2 = ±1. x = -2 ± 1. - 10Solve x² + 6x + 4 = 0 by completing the square, giving your answers in surd form.[3 marks]
Show the answer
x = -3 ± √5
(x + 3)² - 9 + 4 = 0, so (x + 3)² = 5 and x + 3 = ±√5. - 11Solve x² - 2x - 7 = 0 by completing the square, giving your answers to 2 decimal places.[3 marks]
Show the answer
x = 3.83 or x = -1.83
(x - 1)² - 1 - 7 = 0, so (x - 1)² = 8 and x = 1 ± √8.
D · Quadratic formula (Higher)
- 12Solve 2x² + 4x - 1 = 0, giving your answers to 2 decimal places.[3 marks]
Show the answer
x = 0.22 or x = -2.22
a = 2, b = 4, c = -1: x = (-4 ± √24)/4. - 13Solve 3x² - 6x + 2 = 0 using the quadratic formula.[3 marks]
Show the answer
x = 1.58 or x = 0.42
a = 3, b = -6, c = 2: x = (6 ± √12)/6. - 14Solve x² = 3x + 5, giving your answers to 2 decimal places.[3 marks]
Show the answer
x = 4.19 or x = -1.19
Rearrange first: x² - 3x - 5 = 0. a = 1, b = -3, c = -5: x = (3 ± √29)/2. - 15Solve 2x² + 5x - 3 = 0 using the quadratic formula.[3 marks]
Show the answer
x = 0.5 or x = -3
a = 2, b = 5, c = -3: x = (-5 ± √49)/4 = (-5 ± 7)/4. - 16Solve 3x² + 6x + 2 = 0 using the quadratic formula.[3 marks]
Show the answer
x = -0.42 or x = -1.58
a = 3, b = 6, c = 2. x = [-6 ± √(6² - 4×3×2)] / (2×3). x = [-6 ± √(36 - 24)] / 6. x = [-6 ± √12] / 6. x = [-6 ± 2√3] / 6. x = -1 ± (√3)/3. x ≈ -1 ± 0.577. x ≈ -1.58 or -0.42.
E · Choose and explain (Higher)
- 17A rectangle has length (x + 2) and width (x - 1). Its area is 24. Find x.[4 marks]
Show the answer
x = 4.62
x² + x - 26 = 0 gives x = (-1 ± √105)/2. The width x - 1 must be positive, so x = 4.62. - 18For the equation 2x² + 8x + 5 = 0, find the number of real solutions.[4 marks]
Show the answer
2 real solutions
Use discriminant b² - 4ac = 8² - 4×2×5 = 64 - 40 = 24. Since discriminant is positive, there are 2 real solutions. - 19Show that the solutions of 2x² + 5x - 3 = 0 are x = 1/2 and x = -3.[4 marks]
Show the answer
x = 1/2 and x = -3
Use quadratic formula: x = [-5 ± √(25 + 24)] / 4 = [-5 ± √49] / 4 = [-5 ± 7] / 4. So x = (2)/4 = 1/2 or x = (-12)/4 = -3. - 20Show that x² + 4x + 7 = 0 has no real solutions.[4 marks]
Show the answer
b² - 4ac = -12, which is negative, so no real solutions
a = 1, b = 4, c = 7: 4² - 4 × 1 × 7 = 16 - 28 = -12 < 0.
Completing the square and the quadratic formula exam checklist
- Rearrange to = 0 before starting.
- Write a, b, c with their signs included.
- Use brackets around negative numbers for clarity.
- Keep ± until the end to avoid mistakes.
- Round only at the end to keep accuracy.
- Check b² - 4ac to see how many solutions there are.
- Always check your answers by substitution.
Got stuck on a question?
Find the part of this guide it belongs to and go through the worked example again, line by line. Then put your answer back into the question: if it doesn't work, look for a sign or a step you skipped.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
Download the PDF (652 KB)Want maths to click like this?
Every Bloom lesson works the same way: one idea at a time, shown with a picture before it's explained, then practised until it sticks. Book a free assessment and see it applied to your child's own work.
