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.

Step-by-step solution

Idea: Think of the tile rectangle of sides px + a and qx + b: p × q big squares, a columns of q x-tiles and b rows of p x-tiles, and an a × b corner of unit tiles. For example, (2x + 3)(3x + 1) = 6x² + 11x + 3 has 2 × 3 = 6, 2 × 1 + 3 × 3 = 11, 3 × 1 = 3.

1. x² term: px × qx = pqx², so the first blank is pq.½ mark
2. x terms: px × b = pbx and a × qx = qax, so the second blank is pb + qa.½ mark
3. Constant: a × b, so the third blank is ab.½ mark
4. Verify with the distributive property:
   (px + a)(qx + b) = px(qx + b) + a(qx + b) = pqx² + pbx + aqx + ab = (pq)x² + (pb + qa)x + ab ✓½ mark

Check: p = 2, a = 3, q = 3, b = 1: (pq) = 6, (pb + qa) = 2 + 9 = 11, ab = 3, giving 6x² + 11x + 3 = (2x + 3)(3x + 1) ✓. With p = q = 1 it becomes x² + (a + b)x + ab.

Answer to write in the exam

Common mistakes that cost marks

• Writing the middle coefficient as pa + qb. Each letter multiplies the number in the other bracket: p with b, q with a.
• Writing the middle coefficient as a + b, which is only right when p = q = 1.
• Forgetting the pq in front of x².

How this can come in the exam

Try one yourself

Use the result to expand (3x + 2)(5x + 7).

More questions like this

• 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.
• Fill in the blanks to complete the following identities:
• Select and use the identity that will help you to find the following products without multiplying directly:

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