Chapter 8, Problem 58SGR

To determine

To calculate the dimensions of the canvas.

Answer to Problem 58SGR

Dimensions: $(x-4)$ , $(x+6)$

Explanation of Solution

Given:

Area of the canvas $=x^2+2x-24$

Formula used:

Area of the rectangle $=l\times b$

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:

To find the dimensions of the canvas, factorise the given polynomial.

The polynomial given is $x^2+2x-24$ where $a=1,b=2,c=-24$

Substituting in the above formula,

  $\begin{array}{l}\Rightarrow x=\frac{-2\pm \sqrt{2^2-4(1)(-24)}}{2(1)}\\ \Rightarrow x=\frac{-2\pm \sqrt{4+96}}{2}\\ \Rightarrow x=\frac{-2\pm \sqrt{100}}{2}\\ \Rightarrow x=\frac{-2\pm 10}{2}\end{array}$

The roots are,

  $\begin{array}{l}\Rightarrow x_1=\frac{-2+10}{2}=\frac{8}{2}=4\\ \Rightarrow x_2=\frac{-2-10}{2}=\frac{-12}{2}=-6\end{array}$

So,

  $\begin{array}{l}\Rightarrow x^2+2x-24\\ \Rightarrow (x-4)(x-(-6))\\ \Rightarrow (x-4)(x+6)\end{array}$

Conclusion:

Therefore, the dimensions of the canvas are $(x-4)$ and $(x+6).$