Question

How do you factor the expression $ 48{{g}^{2}}-22gh-15{{h}^{2}}$ ?

Answer 1

Hint: In order to do this question, you need to know about factors. If suppose x = i is a factor of the polynomial, then we can say that the x-i is one of the factors of the polynomial. In this question we can consider h as a constant and g as the variable. Then we can factorise the above expression using formula for quadratic equation to get the values of g.

Complete step by step solution:
We are asked to factor the expression $ 48{{g}^{2}}-22gh-15{{h}^{2}}$ .
If we take h as constant and g as variable, we get that the above expression is a quadratic expression in g. We have different methods to obtain the values of g here - factorisation, formula, completing the square method. If we get the values of g, we can express the given equation as factors. So, we will use the formula method.
Therefore, we can factor the given expression as the following:
$ \Rightarrow 48{{g}^{2}}-22gh-15{{h}^{2}}$
Then, we use the formula which we use to find the roots of the quadratic expression. That is :
$ g=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Here, $ a=48,b=-22h,c=-15{{h}^{2}}$. Substituting all the values, we get:
$ \Rightarrow g=\dfrac{22h\pm \sqrt{484{{h}^{2}}+4\times 48\times 15{{h}^{2}}}}{96}$
$ \Rightarrow g=\dfrac{22h\pm \sqrt{484{{h}^{2}}+2880{{h}^{2}}}}{96}$
Simplifying further, we get
$ \Rightarrow g=\dfrac{22h\pm \sqrt{3364{{h}^{2}}}}{96}$
$ \Rightarrow g=\dfrac{22h\pm 58h}{96}$
$ \Rightarrow g=\dfrac{5}{6}h$ or $ \Rightarrow g=\dfrac{-3}{8}h$
Therefore, we can factorize the equation as $ \Rightarrow 48{{g}^{2}}-22gh-15{{h}^{2}}=\left( g-\dfrac{5}{6}h \right)\left( g+\dfrac{3}{8}h \right)$

Note: To do this question, you need to know the formula to find the root of the quadratic expression. Also, you can find if the roots of the equation are real, equal or if they are complex by using the formula of the discriminant which is $ {{b}^{2}}-4ac$. You can also use any other method to get the values of g.