Chapter 1.3, Problem 29AYU

In Problems 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$3x^2=4x+4$

To determine

To find: The solution for the equation $3x^2\text{ = 4}x\text{ + 4}$ algebraically and verify using a graphing utility.

Answer to Problem 29AYU

Solution:

$x\text{ = 2, }\frac{-2}{3}$

Explanation of Solution

Given:

The equation $3x^2\text{ = 4}x\text{ + 4}$

Algebraic solution:

$3x^2\text{ = 4}x\text{ + 4}$

$3x^2\text{ − 4}x\text{ = 4}x\text{ + 4 − 4}x$Subtract $\text{4}x$ from both sides

$3x^2\text{ − 4}x\text{ − 4 = 4 − 4}$Subtract 4 from both sides

$3x^2\text{ − 4}x\text{ − 4 = 0}$

$3x^2\text{ − 6}x\text{ + 2}x\text{ − 4 = 0}$Splitting the middle term

$\text{3}x\left(x\text{-2}\right)\text{ + 2}\left(x\text{-2}\right)\text{ = 0}$Taking $\text{3}x$ common from first two terms and 2 common from last two terms

$\left(x\text{ - 2}\right)\left(\text{3}x\text{ + 2}\right)\text{ = 0}$Taking $\left(x\text{ - 2}\right)$ common factor from the terms

$x\text{ − 2 = 0 or }\left(\text{3}x\text{ + 2}\right)\text{ = 0 }$

$x\text{ − 2 + 2 = 0 + 2 3}x\text{ + 2 − 2 = 0 − 2 }$

$x\text{ = 2 3}x\text{ = −2}$

$x = \frac{-2}{3}$

Check:

Let $x\text{ = 2}$ in the expression in $x$ on the left side of the equation

$3{(2)}^2 = 3\times 4=12$

Let $\text{x = 2}$ in the expression in $x$ on the right side of the equation

$4\left(2\right)+4=8+4$

$=12$

Since the expressions are equal, the solution $x\text{ = 2}$ checks

Let $x$ $= \frac{-2}{3}$ in the expression in $x$ on the left side of the equation

$3{(\frac{-2}{3})}^2 = 3 \times \frac{4}{9}$

$= \frac{4}{3}$

Let $x$ $= \frac{-2}{3}$ in the expression in $x$ on the right side of the equation

$4(\frac{-2}{3}) + 4 = \frac{-8}{3} + 4$

$= \frac{-8+12}{3}$Take LCM

$= \frac{4}{3}$

Since the two expressions are equal, the solution $x = \frac{-2}{3}$ checks.