Chapter 9.5, Problem 78E

To determine

To find: The derivation of a formula that can be used to find solutions of equations that have the form $a{x}^2+x+c=0$ . Also, find the solution of the equation $-2x^2+x+8=0$ .

Answer to Problem 78E

The required formula is $x=\frac{-1\pm \sqrt{1-4ac}}{2a}$ and the required solutions of the equation are $-1.77$ and 2.27 respectively.

Explanation of Solution

The given form of the equation is $a{x}^2+x+c=0$ .

In the above equation, the value of b is 1.

Write the formula of the given equation using the Quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

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

Now, the given equation is $-2x^2+x+8=0$ .

To solve the above equation, use the formula $x=\frac{-1\pm \sqrt{1-4ac}}{2a}$ .

  $\begin{array}{l}x=\frac{-1\pm \sqrt{1-4\left(-2\right)\left(8\right)}}{2\left(-2\right)}\\ x=\frac{-1\pm \sqrt{1+64}}{-4}\\ x=\frac{-1\pm \sqrt{65}}{-4}\end{array}$

Solve the above expression for x .

  $\begin{array}{l}x=\frac{-1+\sqrt{65}}{-4}\\ x=\frac{1-8.06}{4}\\ x\approx -1.77\end{array}$

And, $\begin{array}{l}x=\frac{-1-\sqrt{65}}{-4}\\ x=\frac{1+8.06}{4}\\ x\approx 2.27\end{array}$

Hence, the required formula is $x=\frac{-1\pm \sqrt{1-4ac}}{2a}$ and the required solutions of the equation are $-1.77$ and 2.27 respectively.