ASVAB Study Guide1.Difference of Squares
Factoring a binomial where two perfect squares are being subtracted.
$$a^2 - b^2 = (a - b)(a + b)$$
To factor, take the square root of each term. One binomial gets a minus sign, and the other gets a plus sign.
Example
Factor:
$$x^2 - 25$$
Show solution
- Step 1: Identify terms: $$a^2 = x^2 \Rightarrow a = x$$; $$b^2 = 25 \Rightarrow b = 5$$.
- Step 2: Apply formula $$a^2 - b^2 = (a - b)(a + b)$$.
- Step 3: Write result: $$(x - 5)(x + 5)$$.
=> Conclusion: $$x^2 - 25 = (x - 5)(x + 5)$$
Practice problems
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Problem 1
Factor:
$$x^2 - 49$$
Show solution
Answer: $$(x - 7)(x + 7)$$
- Step 1: $$a = x$$, $$b = \sqrt{49} = 7$$.
- Step 2: Apply formula: $$(x - 7)(x + 7)$$.
=> Conclusion: $$(x - 7)(x + 7)$$.
=> Correct Option: A ($$(x - 7)(x + 7)$$)
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Problem 2
Factor:
$$y^2 - 144$$
Show solution
Answer: $$(y - 12)(y + 12)$$
- Step 1: $$a = y$$, $$b = \sqrt{144} = 12$$.
- Step 2: Apply formula: $$(y - 12)(y + 12)$$.
=> Conclusion: $$(y - 12)(y + 12)$$.
=> Correct Option: B ($$(y - 12)(y + 12)$$)
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Problem 3
Factor:
$$100 - m^2$$
Show solution
Answer: $$(10 - m)(10 + m)$$
- Step 1: $$a = \sqrt{100} = 10$$, $$b = m$$.
- Step 2: Apply formula: $$(10 - m)(10 + m)$$.
=> Conclusion: $$(10 - m)(10 + m)$$.
=> Correct Option: A ($$(10 - m)(10 + m)$$)