Chapter 6.8, Problem 50HP

To determine

To calculate; the roots of $x^2+bx+c=0$ are $M$ and $N$ . If $|M-N|=1$ , express $c$ in terms of $b.$

Answer to Problem 50HP

Solution: $c=\frac{b^2-1}{4}$

Explanation of Solution

Using the quadratic formula, the roots are:

  $\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\frac{-b\pm \sqrt{b^2-4(1)c}}{2(1)}\\ =\frac{-b\pm \sqrt{b^2-4c}}{2}\end{array}$

Given equation

  $|M-N|=1$

Substitute the values of $M$ and $N$

  $\begin{array}{l}|\frac{-b+\sqrt{b^2-4c}}{2}-\frac{-b-\sqrt{b^2-4c}}{2}|=1\\ \left|\frac{-2\sqrt{b^2-4c}}{2}\right|=1\\ |-\sqrt{b^2-4c}|=1\\ \sqrt{b^2-4c}=1\\ $ {(\sqrt{b^2-4c})}^2=1^2$b^2-4c=1\\ b^2-4c-b^2=1-b^2\end{array}$

  $\begin{array}{l}-4c=1-b^2\\ \frac{-4c}{-4}=\frac{1-b^2}{-4}\\ c=\frac{b^2-1}{4}\end{array}$