How to Factor x^2 + bx + c

Step-by-step solution

1. List number pairs that multiply to 12.

   $$1 \cdot 12,\quad 2 \cdot 6,\quad 3 \cdot 4$$

   Why: The constant term 12 comes from multiplying the two numbers in the brackets.

2. Find the pair that adds to 7.

   $$3 + 4 = 7$$

   Why: The middle coefficient 7 is the sum of the two numbers.

3. Write the factored form with the pair.

   $$(x + 3)(x + 4)$$

   Why: Both signs are plus because 12 and 7 are both positive.

4. Check by expanding.

   $$(x + 3)(x + 4) = x^2 + 7x + 12$$

   Why: Expanding returns the original trinomial.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $x^2 + 5x + 6$

   Show answer

   $(x + 2)(x + 3)$

2. 2

   $x^2 - x - 6$

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   $(x - 3)(x + 2)$

3. 3

   $x^2 - 9x + 20$

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   $(x - 4)(x - 5)$

4. 4

   $x^2 + 2x - 15$

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   $(x + 5)(x - 3)$

5. 5

   $x^2 - 11x + 24$

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   $(x - 3)(x - 8)$