Solve by factorising:
x2 - 8x + 15 = 0
x2 - 8x + 15 = 0
Worked Solution & Answer
(x - 3)(x - 5) = 0 ⇒ x = 3 or x = 5 B1
Answer: x = 3, x = 5
Solving quadratic equations — Notes & Worked Examples Show / Hide
Explanation:
To solve a quadratic equation by factorising, first rearrange the equation so that one side equals zero (ax2 + bx + c = 0). Never factorise while there is a non-zero number on the right-hand side.
Factorise the quadratic expression into two brackets, then set each bracket equal to 0 and solve for x (because if two brackets multiply to give 0, at least one bracket must equal 0). Notice that the signs of the roots are opposite to the signs in the brackets.
Example 1: Solve x2 − 8x + 15 = 0.
Solution:
Multiply to +15 and add to −8 → −3 and −5
Factorise: (x − 3)(x − 5) = 0
Either x − 3 = 0 or x − 5 = 0
Answer: x = 3 or x = 5
Example 2: Solve x2 + 3x = 18.
Solution:
Rearrange to equal 0: x2 + 3x − 18 = 0
Multiply to −18, add to +3 → +6 and −3
Factorise: (x + 6)(x − 3) = 0
x + 6 = 0 → x = −6; x − 3 = 0 → x = 3
Answer: x = −6 or x = 3
Solve by factorising:
x2 - 7x + 12 = 0
x2 - 7x + 12 = 0
Worked Solution & Answer
(x - 3)(x - 4) = 0 ⇒ x = 3 or x = 4 B1
Answer: x = 3, x = 4
Solve (x - 2)2 = 3
Give your solutions correct to 3 significant figures.
Give your solutions correct to 3 significant figures.
Worked Solution & Answer
x – 2 = ±√3 M1
⇒ x = 2 ± 1.73205...
⇒ x = 3.73 or x = 0.268 A1
⇒ x = 2 ± 1.73205...
⇒ x = 3.73 or x = 0.268 A1
Answer: 3.73, 0.268
Solve x2 = 5x + 24
Worked Solution & Answer
x² – 5x – 24 = 0 M1
⇒ (x – 8)(x + 3) = 0 M1
⇒ x = 8 or x = –3 A1
⇒ (x – 8)(x + 3) = 0 M1
⇒ x = 8 or x = –3 A1
Answer: x = 8 and x = -3
Solve x2 - 2x - 15 = 0
Worked Solution & Answer
(x – 5)(x + 3) = 0 M2
⇒ x = 5 or x = –3 A1
⇒ x = 5 or x = –3 A1
Answer: x = 5, x = –3
Solve x2 - 7x - 18 = 0
Worked Solution & Answer
(x + 2)(x – 9) = 0 M2
⇒ x = –2 or x = 9 A1
⇒ x = –2 or x = 9 A1
Answer: x = -2,\ x = 9
Solve x2 + 5x - 24 = 0
Worked Solution & Answer
(x + 8)(x – 3) = 0 M2
⇒ x = –8 or x = 3 A1
⇒ x = –8 or x = 3 A1
Answer: x = -8,\ x = 3
Solve 6x2 + 5x - 6 = 0
Worked Solution & Answer
(2x + 3)(3x – 2) = 0 M2
⇒ x = –3/2 (or –1.5) and x = 2/3 A1
⇒ x = –3/2 (or –1.5) and x = 2/3 A1
Answer: -32 and 23
Rectangle with sides (x + 2) cm and (x – 5) cm has area 120 cm². Work out x. [4 marks]
The area of the rectangle is 120 cm2
Work out the value of x.
Work out the value of x.
x =
Worked Solution & Answer
(The area of a rectangle is length × width)
(x + 2)(x – 5) = 120 ⇒ x² – 3x – 10 = 120 ⇒ x² – 3x – 130 = 0. M3
Factorise: (x – 13)(x + 10) = 0 ⇒ x = 13 (length > 0). A1
(x + 2)(x – 5) = 120 ⇒ x² – 3x – 10 = 120 ⇒ x² – 3x – 130 = 0. M3
Factorise: (x – 13)(x + 10) = 0 ⇒ x = 13 (length > 0). A1
Answer: x = 13
a (a) Complete the table for y = x2 - 2x.
@@TABULAR0@@
b (b) Draw the graph of y = x2 - 2x for -1 ≤ x ≤ 4.
c (c) Use your graph to solve x2 - 2x = 2.
@@TABULAR0@@
b (b) Draw the graph of y = x2 - 2x for -1 ≤ x ≤ 4.
c (c) Use your graph to solve x2 - 2x = 2.
(a) Complete the table for y = x2 - 2x.
@@TABULAR0@@ [1 mark]
@@TABULAR0@@ [1 mark]
(b) Draw the graph of y = x2 - 2x for -1 ≤ x ≤ 4. [2 marks]
(c) Use your graph to solve x2 - 2x = 2. [2 marks]
Worked Solution & Answer
(a)
When x = 4, y = 42 - 2(4) = 16 - 8 = 8 B1
Answer (a): 8
(b)
Plot the points and join with a smooth parabola M1 A1
Answer (b): Smooth parabola plotted through (−1, 3), (0, 0), (1, −1), (2, 0), (3, 3), (4, 8)
(c)
Draw the horizontal line y = 2 M1
it intersects the curve at x ≈ -0.7 and x ≈ 2.7 A1
it intersects the curve at x ≈ -0.7 and x ≈ 2.7 A1
Answer (c): x = -0.7 and x = 2.7