We have seen that (x + 3)(x + 4) = x2 + 7x + 12.
Also (x + 6)(x + 7) = x2 + 13x + 42.
Generalise the pattern to get an expression for (x + a) (x + b).
Answer: (x + a)(x + b) = x2 + (a + b)x + ab: the coefficient of x is the sum of the two numbers and the constant is their product.
Step-by-step solution
Given: (x + 3)(x + 4) = x2 + 7x + 12; (x + 6)(x + 7) = x2 + 13x + 42
To find: A general expression for (x + a)(x + b)
To find: A general expression for (x + a)(x + b)
Idea: Look at where 7, 12, 13 and 42 come from: 7 = 3 + 4 and 12 = 3 × 4; 13 = 6 + 7 and 42 = 6 × 7. Guess the rule, then prove it with the distributive property.
- Spot the pattern: in (x + 3)(x + 4), 3 + 4 = 7 (the x-coefficient) and 3 × 4 = 12 (the constant). In (x + 6)(x + 7), 6 + 7 = 13 and 6 × 7 = 42.½ mark
- Guess: (x + a)(x + b) = x2 + (a + b)x + ab.½ mark
- Prove it: (x + a)(x + b) = x(x + b) + a(x + b) = x2 + bx + ax + ab½ mark
- = x2 + (a + b)x + ab. This holds for all values, so it is an identity.½ mark
(x + a)(x + b) = x² + (a + b)x + ab.
Check: a = 3, b = 4: x2 + 7x + 12 ✓. a = 5, b = −2: (x + 5)(x − 2) = x2 + 3x − 10, and by the formula (5 − 2) = 3, 5 × (−2) = −10 ✓.
Answer to write in the exam
(x + 3)(x + 4) = x2 + (3 + 4)x + 3 × 4; (x + 6)(x + 7) = x2 + (6 + 7)x + 6 × 7
(x + a)(x + b) = x(x + b) + a(x + b)
= x2 + bx + ax + ab
∴ (x + a)(x + b) = x2 + (a + b)x + ab
Common mistakes that cost marks
- Writing the middle term as abx. The coefficient of x is the sum a + b, not the product.
- Writing the constant as a + b. The constant is the product ab.
- Losing signs when a or b is negative, e.g. (x − 3)(x − 4) = x2 − 7x + 12 (the constant is + 12 because (−3)(−4) = 12).
How this can come in the exam
MCQ (1 mark)
(x + 8)(x − 3) equals
- x2 + 5x − 24
- x2 − 5x − 24
- x2 + 11x − 24
- x2 + 5x + 24
Show answer
(A) x2 + 5x − 24
a + b = 8 + (−3) = 5 and ab = 8 × (−3) = −24.
Short answer (2 marks)
Use (x + a)(x + b) = x2 + (a + b)x + ab to find 103 × 104.
Show answer
Take x = 100, a = 3, b = 4 (1 mark): 1002 + 7 × 100 + 12 = 10000 + 700 + 12 = 10712 (1 mark).Try one yourself
Find (y + 9)(y + 5) using the pattern.
Show answer
y2 + (9 + 5)y + 9 × 5 = y2 + 14y + 45.
More questions like this
- Now consider the case where we have a rectangle of sidelengths 2x + 3 and 3x + 1, as shown in the figure. What can you say about its area (2x + 3) (3x + 1)?
- Fill in the blanks with the appropriate expressions to make the equation true.
(px + a) (qx + b) = (_____)x2 + (_____)x + _____ .
Also, verify your answer using the distributive property. - Let us begin with x2 + 7x + 12 = x2 + (a + b)x + ab.
- Let us try to factor x2 + 11x + 30 in a similar manner.
- In order to factor x2 − 5x + 6, we first note that the coefficient of x is negative.