Chapter 8.7, Problem 61PFA

To determine

To calculate the dimensions of the swimming pool.

Answer to Problem 61PFA

  $3x(2x-7)(2x-7)$

Explanation of Solution

Given:

Volume of the swimming pool is $12x^3-84x^2+147x$

Formula used:

Roots of the polynomial of the form $a{x}^2+bx+c=a(x-x_1)(x-x_2)$ are

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Consider the expression,

  $\Rightarrow 12x^3-84x^2+147x$

Finding the common terms in the expression,

  $\Rightarrow x(12x^2-84x+147)$

Applying the above formula for $a=12,b=-84,c=147$

  $\begin{array}{l}\Rightarrow x=\frac{-(-84)\pm \sqrt{{(-84)}^2-4(12)(147)}}{2(12)}\\ \Rightarrow x=\frac{84\pm \sqrt{7056-7056)}}{24}\\ \Rightarrow x=\frac{84}{24}=\frac{21}{6}=\frac{7}{2}\end{array}$

Therefore, the roots are

  $\Rightarrow x_1=x_2=\frac{7}{2}$

Thus, $12x^2-84x+147=3(2x-7)(2x-7).$

Conclusion:

Therefore, the given expression is factorised as $3x(2x-7)(2x-7).$