Chapter EP, Problem 5.2.11EP

To determine

To find: the solution of the given quadratic equation. If exact root can not be found find the two consecutive integers between which the roots are located.

Answer to Problem 5.2.11EP

The roots are located between two integers $-1$ and 2.

Explanation of Solution

Given information:

The quadratic equation provided is

  $4x+1=3x^2$

Concept used:

Solution of a quadratic equation can be found by factorizing the equation and equating each product with zero.

Calculation:

Consider the given equation.

  $\begin{array}{l}4x+1=3x^2\\ 3x^2-4x-1=0\end{array}$

Now, use quadratic formula to solve the equation as shown:

  $\begin{array}{c}3x^2-4x-1=0\\ x=\frac{4\pm \sqrt{4^2-4\left(3\right)\left(-1\right)}}{2\cdot 3}\\ =\frac{4\pm \sqrt{16+12}}{6}\\ =\frac{4\pm \sqrt{28}}{6}\\ =\frac{4\pm 2\sqrt{7}}{6}\\ =\frac{2\pm \sqrt{7}}{3}\end{array}$

The exact root can not be found.

Now,

  $\begin{array}{c}4<7<9\\ \sqrt{4}<\sqrt{7}<\sqrt{9}\\ 2<\sqrt{7}<3\\ 2+2<\sqrt{7}+2<3+2\\ \frac{4}{3}<\frac{\sqrt{7}+2}{3}<\frac{5}{3}\end{array}$

The roots are located between two integers $-1$ and 2.