EDEXCEL GCSE MATHS · HIGHER
Solving quadratic equations 2
Equations and inequalities · Lesson 3 of 7
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
1. Solve x² − 5x + 6 = 0.
x = 2 or x = 3
2. Expand (2x + 1)(x + 3).
2x² + 7x + 3
3. Work out √81.
9
4. Work out 5² − 4 × 1 × 6.
1
5. Round 5.385 to 2 decimal places.
5.39
Learning Objectives
1. Factorise and solve ax² + bx + c = 0.
2. Use the quadratic formula.
3. Give answers to a given accuracy or as surds.
4. Decide which method to use.
Factorising ax² + bx + c
Use the product ac and the sum b.
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1 Multiply a by c For 2x² + 7x + 3: 2 × 3 = 6 |
2 Find two numbers with that product and sum b 1 and 6 (they add to 7) |
3 Split the middle term 2x² + x + 6x + 3 |
4 Factorise in pairs x(2x + 1) + 3(2x + 1) |
5 Write as two brackets (2x + 1)(x + 3) |
A Harder Factorisation
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Solve 2x² + 7x + 3 = 0. |
1. Factorise
(2x + 1)(x + 3) = 0
2. Set each bracket to zero
2x + 1 = 0 or x + 3 = 0
3. Solve
x = −½ or x = −3
Answer: x = −½ or x = −3
Another Factorisation
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Solve 3x² − 10x + 8 = 0. |
1. ac = 24; two numbers with product 24 and sum −10
−4 and −6
2. Split and factorise
3x² − 4x − 6x + 8 = x(3x − 4) − 2(3x − 4)
3. Write as brackets
(3x − 4)(x − 2) = 0
4. Solve
x = 4/3 or x = 2
Answer: x = 4/3 or x = 2
The Quadratic Formula
For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac))/2a.
Learn it: it is not on the formulae sheet.
When It Will Not Factorise
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The roots are not whole numbers, so factorising will not work. |
The graph of y equals x squared plus 3x minus 5 crossing the x-axis at about minus 4.19 and 1.19.
Using the Formula
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Solve x² + 3x − 5 = 0, giving your answers to 2 decimal places. |
1. Identify a = 1, b = 3, c = −5
Substitute into the formula
2. b² − 4ac
9 − 4(1)(−5) = 9 + 20 = 29
3. Write the formula
x = (−3 ± √29)/2
4. Calculate
√29 = 5.385...
5. The two solutions
x = 1.19 or x = −4.19
Answer: x = 1.19 or x = −4.19
Leaving the Answer as a Surd
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Solve x² − 6x + 2 = 0, giving your answers in the form p ± √q. |
1. a = 1, b = −6, c = 2
b² − 4ac = 36 − 8 = 28
2. Formula
x = (6 ± √28)/2
3. Simplify √28 = 2√7
x = (6 ± 2√7)/2
4. Divide through by 2
x = 3 ± √7
Answer: x = 3 + √7 or x = 3 − √7
Which Method?
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FACTORISING |
THE QUADRATIC FORMULA |
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▸ Quick when the numbers are simple. ▸ Works when the roots are rational. ▸ Gives exact answers. |
▸ Always works. ▸ Needed for "give to 2 decimal places" or surds. ▸ Take care with the signs of b and c. |
Common Mistakes
The formula has plenty of places to slip.
▸ Sign of b. The formula starts with −b; if b = −6, then −b = 6.
▸ The bracket. b² must be squared before any minus: (−6)² = 36.
▸ Dividing. Divide the WHOLE numerator by 2a, not just the surd.
▸ Order of the calculation. Use brackets on the calculator for the top line.
Key Terms
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Coefficient The number multiplying a variable, such as the 2 in 2x². |
Quadratic formula x = (−b ± √(b² − 4ac))/2a. |
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Discriminant The expression b² − 4ac under the square root. |
Surd A root that cannot be simplified to a whole number, such as √7. |
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Root A solution of an equation. |
Substitute Replace letters with numbers. |
Your Task: Choose Your Method
12 minutes
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For each equation, decide whether to factorise or use the formula, then solve it. (a) 2x² − 5x − 12 = 0 (b) x² + 4x − 3 = 0 (to 2 d.p.) (c) x² − 10x + 25 = 0. 1. Test for simple factors. 2. Otherwise use the formula. |
A good answer shows: (a) (2x + 3)(x − 4) = 0: x = −1.5 or 4. (b) Formula: (−4 ± √28)/2 = −2 ± √7: x = 0.65 or −4.65. (c) (x − 5)² = 0: x = 5 (one repeated solution).
Note: Ask what the discriminant tells you in (c).
Can I...?
☐ Factorise ax² + bx + c.
☐ Solve by factorising.
☐ Write down the quadratic formula.
☐ Identify a, b and c correctly.
☐ Substitute carefully with brackets.
☐ Give answers to 2 decimal places.
☐ Give answers as surds.
☐ Choose between methods.
Summary
✓ x = (−b ± √(b² − 4ac))/2a.
✓ Factorise when the numbers are simple; otherwise use the formula.
✓ Watch the signs and divide the whole numerator by 2a.
✓ Check by substituting your answers into the original equation.
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EXAM FOCUS Solve x² + 3x − 5 = 0. Give your solutions correct to 2 decimal places. (3 marks) Write a, b and c first, and put brackets round negative numbers when you substitute. |
Exam Practice: Solving Quadratic Equations 2
Answer all questions. Show your working. · 30 minutes
▸ Question 1 · 3 marks · Non-calculator. Solve 2x² + 7x + 3 = 0.
▸ Question 2 · 3 marks · Non-calculator. Solve 3x² − 10x + 8 = 0.
▸ Question 3 · 4 marks · Non-calculator. Solve 2x² − 5x − 12 = 0.
▸ Question 4 · 3 marks · Calculator. Solve x² + 3x − 5 = 0. Give your solutions correct to 2 decimal places.
▸ Question 5 · 3 marks · Calculator. Solve 3x² − 4x − 2 = 0. Give your solutions correct to 2 decimal places.
▸ Question 6 · 4 marks · Non-calculator. Solve x² − 6x + 2 = 0. Give your answers in the form p ± √q, where p and q are integers.
Question 1 · 3 marks · Non-calculator
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“Solve 2x² + 7x + 3 = 0.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 1 · mark scheme
3 marks available. Award a mark for each point made.
▸ Correct factorisation. M1
▸ One solution. A1
▸ Both solutions. A1
▸ Model answer. (2x + 1)(x + 3) = 0, so x = −½ or x = −3.
Question 2 · 3 marks · Non-calculator
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“Solve 3x² − 10x + 8 = 0.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 2 · mark scheme
3 marks available. Award a mark for each point made.
▸ Correct factorisation. M1
▸ One solution. A1
▸ Both solutions. A1
▸ Model answer. (3x − 4)(x − 2) = 0, so x = 4/3 or x = 2.
Question 3 · 4 marks · Non-calculator
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“Solve 2x² − 5x − 12 = 0.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 4 marks, so plan before writing.
Question 3 · mark scheme
4 marks available. Award a mark for each point made.
▸ 2x² − 8x + 3x − 12. M1
▸ (2x + 3)(x − 4). M1
▸ One solution. A1
▸ Both solutions. A1
▸ Model answer. (2x + 3)(x − 4) = 0, so x = −3/2 or x = 4.
Question 4 · 3 marks · Calculator
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“Solve x² + 3x − 5 = 0. Give your solutions correct to 2 decimal places.” |
HOW TO ANSWER IT Command word: Calculator. Worth 3 marks, so plan before writing.
Question 4 · mark scheme
3 marks available. Award a mark for each point made.
▸ Correct substitution into the formula. M1
▸ (−3 ± √29)/2. A1
▸ 1.19 and −4.19. A1
▸ Model answer. x = (−3 ± √(9 + 20))/2 = (−3 ± √29)/2, so x = 1.19 or x = −4.19.
Question 5 · 3 marks · Calculator
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“Solve 3x² − 4x − 2 = 0. Give your solutions correct to 2 decimal places.” |
HOW TO ANSWER IT Command word: Calculator. Worth 3 marks, so plan before writing.
Question 5 · mark scheme
3 marks available. Award a mark for each point made.
▸ Correct substitution. M1
▸ (4 ± √40)/6. A1
▸ 1.72 and −0.39. A1
▸ Model answer. x = (4 ± √(16 + 24))/6 = (4 ± √40)/6, so x = 1.72 or x = −0.39.
Question 6 · 4 marks · Non-calculator
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“Solve x² − 6x + 2 = 0. Give your answers in the form p ± √q, where p and q are integers.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 4 marks, so plan before writing.
Question 6 · mark scheme
4 marks available. Award a mark for each point made.
▸ Correct substitution. M1
▸ √28. A1
▸ 2√7. M1
▸ 3 ± √7. A1
▸ Model answer. x = (6 ± √(36 − 8))/2 = (6 ± √28)/2 = (6 ± 2√7)/2 = 3 ± √7.