Chapter 9.5, Problem 47PPS

To determine

The solutions of the equation $2.3x^2-1.4x=6.8$ using quadratic formula

Answer to Problem 47PPS

  $x=-0.5+\frac{\sqrt{64.52}}{4.6},\text{and }x=-\text{0}\text{.5}-\frac{\sqrt{64.52}}{4.6}$

Explanation of Solution

Given:

The equation,

  $2.3x^2-1.4x=6.8$

Concept Used:

Quadratic formula: The roots of a quadratic equation $a{x}^2+bx+c=0;\text{ where }a\ne 0$ are:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

In order to solve the given equation $2.3x^2-1.4x-6.8=0$ , since it is in the descending order of exponents, so comparing with the general equation $a{x}^2+bx+c=0$ , the value of the coefficient a , b , and c is given by

  $a=2.3,b=-1.4,\text{ and }c=-6.8$

Now, above quadratic equation $2.3y^2-1.4y-6.8=0$ can be solved using quadratic formula as shown below,

  $\begin{array}{c}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\frac{-\left(1.4\right)\pm \sqrt{{\left(1.4\right)}^2-4\left(2.3\right)\left(-6.8\right)}}{2\left(2.3\right)}\\ =\frac{-2.3\pm \sqrt{1.96+62.56}}{4.6}\\ =\frac{-2.3\pm \sqrt{64.52}}{4.6}\\ x=\frac{-2.3+\sqrt{64.52}}{4.6},\text{and }x=\frac{-2.3-\sqrt{64.52}}{4.6}\\ x=\frac{-2.3}{4.6}+\frac{\sqrt{64.52}}{4.6},\text{and }x=\frac{-2.3}{4.6}-\frac{\sqrt{64.52}}{4.6}\\ x=-0.5+\frac{\sqrt{64.52}}{4.6},\text{and }x=-\text{0}\text{.5}-\frac{\sqrt{64.52}}{4.6}\end{array}$

Thus, the solutions of the given question are, $x=-0.5+\frac{\sqrt{64.52}}{4.6},\text{and }x=-\text{0}\text{.5}-\frac{\sqrt{64.52}}{4.6}$