How do you factor $12 x^{2} + 60 x + 75$ $12x^2 + 60x + 75$ ?

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How do you factor $12 x^{2} + 60 x + 75$ $12x^2 + 60x + 75$ ?

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Answer 1 · 2016-10-30T00:08:54

$12 x^{2} + 60 x + 75 = 3 {(2 x + 5)}^{2}$ $12x^2+60x+75 = 3(2x+5)^2$

Explanation:

Note that all of the terms are divisible by $3$ $3$ , so separate that out as a scalar factor first:

$12 x^{2} + 60 x + 75 = 3 (4 x^{2} + 20 x + 25)$ $12x^2+60x+75 = 3(4x^2+20x+25)$

Next note that $4 x^{2} = {(2 x)}^{2}$ $4x^2 = (2x)^2$ and $25 = 5^{2}$ $25 = 5^2$ are both perfect squares. So do we have a perfect square trinomial?

Let's try:

${(2 x + 5)}^{2} = {(2 x)}^{2} + 2 (2 x) (5) + 5^{2} = 4 x^{2} + 20 x + 25$ $(2x+5)^2 = (2x)^2+2(2x)(5) + 5^2 = 4x^2+20x+25$

...matching our parenthesised quadratic.

So putting it all together we have:

$12 x^{2} + 60 x + 75 = 3 {(2 x + 5)}^{2}$ $12x^2+60x+75 = 3(2x+5)^2$

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How do you factor $12 x^{2} + 60 x + 75$ $12x^2 + 60x + 75$ ?

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