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Question: How do you find the product\[\left( g-4h \right)\left( g-4h \right)\]?...

How do you find the product(g−4h)(g−4h)\left( g-4h \right)\left( g-4h \right)?

Explanation

Solution

In this question, we have to find the product of the two equations which are in the linear form. We will solve questions by using the foil method. The foil method states that if we multiply (a+b) and (c+d), then (a+b)(c+d) will be equal to (ac+ad+bc+bd). After solving this question, we will see the alternate method of solving this question.

Complete answer:
Let us solve this question. We have to find the product of (g-4h)(g-4h).
That means we can say that we have to multiply (g-4h) two times.
Let us find the product of (g-4h)(g-4h).
We will use here foil method to find the product of (g-4h)(g-4h).
According to foil method, we can write the equation as
(g−4h)(g−4h)=g×g+g×(−4h)+(−4h)×g+(−4h)×(−4h)(g-4h)(g-4h)=g\times g+g\times (-4h)+(-4h)\times g+(-4h)\times (-4h)
We can write the above equation as
⇒(g−4h)(g−4h)=(g)2−4gh+(−4gh)+(−4h)2\Rightarrow (g-4h)(g-4h)={{(g)}^{2}}-4gh+\left( -4gh \right)+{{(-4h)}^{2}}
The square of -4 will be 16. So, we can write the above equation as
⇒(g−4h)(g−4h)=(g)2−4gh−4gh+16(h)2\Rightarrow (g-4h)(g-4h)={{(g)}^{2}}-4gh-4gh+16{{(h)}^{2}}
⇒(g−4h)(g−4h)=g2−4gh−4gh+16h2\Rightarrow (g-4h)(g-4h)={{g}^{2}}-4gh-4gh+16{{h}^{2}}
We can write the above equation as
⇒(g−4h)(g−4h)=g2−8gh+16h2\Rightarrow (g-4h)(g-4h)={{g}^{2}}-8gh+16{{h}^{2}}
Hence, we can say that the product of (g−4h)(g−4h)(g-4h)(g-4h) is g2−8gh+16h2{{g}^{2}}-8gh+16{{h}^{2}}.

Note: We can solve this question using an alternate method.
Let us solve this question using the alternate method.
As we can see that we have to find the product of (g-4h)(g-4h).
We can see here that the linear equation (g-4h) has to be multiplied two times.
And we know that if the linear equation (say, ax+b) has to be multiplied two times, then we can write the product as the square of the linear equation as (ax+b)2{{(ax+b)}^{2}}.
So, we can write (g−4h)(g−4h)=(g−4h)2(g-4h)(g-4h)={{\left( g-4h \right)}^{2}}
And we know that (a−b)2=a2−2ab+b2{{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}
Hence, using the above formula, we can write
(g−4h)2=g2−8gh+16h2{{\left( g-4h \right)}^{2}}={{g}^{2}}-8gh+16{{h}^{2}}
Therefore, from both the methods, we are getting the same values. So, we can use this method too.