Chapter 3.1, Problem 64E

To determine

To calculate: .The equation of odd integer whose product is $143$ and solve the equation.

Answer to Problem 64E

The required solution is $11$ and $-13$ .

Explanation of Solution

Given Information:

The product of two consecutive odd numbers is $143$ .

Formula Used:

Quadratic formula:

If $a{x}^2+bx+c=0$ then $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Where $a,b,c$ are constant and $a\ne 0$ .

Calculation:

Consider the product of two consecutive number is $143$ .

Let, one number is $x$ .

Then, other number be $x+2$ .

Now, the equation of two consecutive odd number is:

  $x\left(x+2\right)=143$

Simplify the above equation.

  $\begin{array}{c}x\left(x+2\right)=143\\ x^2+2x=143\\ x^2+2x-143=0\end{array}$

Now, use quadratic formula in the above equation.

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Substitute $1$ for $a$ , $2$ for $b$ and $-143$ for $c$ in the above equation.

  $\begin{array}{c}x=\frac{-2\pm \sqrt{2^2-4\left(1\right)\left(-143\right)}}{2\left(1\right)}\\ =\frac{-2\pm \sqrt{4+572}}{2}\\ =\frac{-2\pm \sqrt{576}}{2}\\ =\frac{-2\pm 24}{2}\end{array}$

Simplify the equation.

  $\begin{array}{c}x=\frac{-2\pm 24}{2}\\ =\frac{-2+24}{2},\frac{-2-24}{2}\\ =\frac{22}{2},\frac{-26}{2}\\ =11,-13\end{array}$

Thus, the solution of $x$ are $11$ and $-13$ .

Hence, the required solution of $x$ are $11$ and $-13$ .