How to simplify $2(6x - 7)^2 + 5$?

Question

How does this simplify?

$$2(6x - 7)^2 + 5 = 2(36^2 - 84x + 49) + 5$$

$6$ squared is $36$, $x$ squared is $x^2$, $-7$ squared is $+49$. Where does $-84x$ come from?

Full problem in question:

Answer 1

The general property is $(a+b)^2=a^2+2ab+b^2$, which you can prove by distributing it out. The $-84x=2 (6x)(-7)$

Answer 2

Perfect Square Trinomial: The expansion of $(ax\pm b)^2$ is $a^2x^2\pm 2abx+b^2$.

This can be proven by rewriting $(ax\pm b)^2$ equal to $(ax\pm b)(ax\pm b)$ and expanding.

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Therefore,$$\begin{align*}(6x-7)^2=6^2x^2-2\cdot6\cdot7x+7^2x^2=36x^2-84x+49\end{align*}$$

Answer 3

$$\begin{align}2(6x-7)^2+5&=2(6x-7)(6x-7)+5\\&=2(6x\cdot6x-7\cdot6x-7\cdot6x+(-7)\cdot(-7))+5\\&=2(36x^2-42\cdot2x+49)+5\\&=2(36x^2-84x+49)+5\end{align}$$ Note that $(6x-7)^2\neq(6x)^2+(-7)^2$ in general (you must use binomial expansion: https://en.wikipedia.org/wiki/Binomial_theorem).

Answer 4

Just remember this acronym: PEMDAS. Since this question is about simplifying an expression, but with numerous operations, PEMDAS is a real good way to remember in what order should you do these calculations. PEMDAS means Parenthesis Exponent Multiplication Division Addition Subtraction. This is typically the order you should appoach and you work left to right. The amswers above are using PEMDAS and this is how you carry out the problem.