Chapter 12, Problem 72CLR

To determine

To solve: The given equation, inequality, or system, assuming that no denominator is zero.

Answer to Problem 72CLR

The solution is $\left\{\frac{-1+\sqrt{22}}{3},\frac{-1-\sqrt{22}}{3}\right\}$ .

Explanation of Solution

Given:

The equation is $3t^2+2t-7=0$ .

Calculation:

For an equation $a{x}^2+bx+c=0$ , the solutions can be determined using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Compare the given equation $3t^2+2t-7=0$ with the standard form $a{x}^2+bx+c=0$ to determine the values of a , b and c .

  $\begin{array}{c}a=3\\ b=2\\ c=-7\end{array}$

Substitute the values of a , b and c in the quadratic formula and solve as shown below:

  $\begin{array}{c}x=\frac{-2\pm \sqrt{2^2-4\left(3\right)\left(-7\right)}}{2\left(3\right)}\\ =\frac{-2\pm \sqrt{4+84}}{6}\\ =\frac{-2\pm \sqrt{88}}{6}\end{array}$

Now, 88 can be written as $4\times 22$

and 4 can be taken out of the root sign, as shown below:

  $\begin{array}{c}x=\frac{-2\pm 2\sqrt{22}}{6}\\ =\frac{-1\pm \sqrt{22}}{3}\\ =\frac{-1+\sqrt{22}}{3},\frac{-1-\sqrt{22}}{3}\end{array}$

So, the solution of the given equation is $\left\{\frac{-1+\sqrt{22}}{3},\frac{-1-\sqrt{22}}{3}\right\}$ .