Chapter 12, Problem 10CT

To determine

To calculate: The values of $x$ from the equation $3x^2-10x-4=0$ solve by using the most appropriate method.

Answer to Problem 10CT

The values are $x$ is $\frac{5\pm \sqrt{7}}{3}$ .

Explanation of Solution

Given information:

The equation is given as:

  $3x^2-10x-4=0$

Formula used:

Methods for solving a Quadratic equation,

If the equation is in this form $a{x}^2+bx+c=0$ then use quadratic method.

If the equation is in this form $a{x}^2+bx=0$ then use Factoring method.

If the equation is in this form $a{x}^2+c=0$ then use property of square roots of equal numbers.

If the equation is in this form $x^2+bx+c=0$ and $\text{b}$ is an even number then use Completing the square method.

Quadratic formula,

If $a{z}^2+bz+c=0$ , $a\ne 0$ and $b^2-4ac\ge 0$ .

Then $z=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Since, the provided equation $3x^2-10x-4=0$ is in the form $a{x}^2+bx+c=0$ so, using the quadratic method as,

Using the quadratic formula,

The values of $a=3,b=-10\text{and}c=-4$ put into the quadratic formula,

  $\begin{array}{l}x=\frac{-\left(-10\right)\pm \sqrt{100-4\cdot 3\cdot \left(-4\right)}}{2\cdot 3}\\ x=\frac{10\pm \sqrt{148}}{6}\\ x=\frac{10\pm 2\sqrt{7}}{6}\end{array}$

Furthermore,

  $\begin{array}{l}x=\frac{5\pm \sqrt{7}}{3}\\ x=\frac{5+\sqrt{7}}{3}\text{or}x=\frac{5-\sqrt{7}}{3}\end{array}$

The value of $x$ is $\frac{5\pm \sqrt{7}}{3}$ .