Chapter 3.5, Problem 55E

To determine

To determine: The two different ways to solve the quadratic equation.

Answer to Problem 55E

The solutions are $x=3,4$ and the factorization method is preferable.

Explanation of Solution

Given information:

The quadratic equation: $-2x^2+12x-17=2x^2-16x+31$

Calculation:

The given equation is $-2x^2+12x-17=2x^2-16x+31$ .

Rewrite the given equation as:

  $\begin{array}{c}-2x^2+12x-17=2x^2-16x+31\\ 2x^2-16x+31+2x^2-12x+17=0\\ \frac{\cancel{4}{x}^2}{\cancel{4}}-\frac{{}^7\cancel{28}x}{\cancel{4}}+\frac{{}^{12}\cancel{48}}{\cancel{4}}=\frac{0}{4}\\ x^2-7x+12=0\end{array}$

The above equation can be solved by two different ways which are graphically or by factorization.

Graphical method:

The graph of the above expression is shown below:

  

From the above graph it is seen that the curve intersects the x -axis at the points $\left(3,0\right)$ and $\left(4,0\right)$ .

Thus, the solutions of the given equation are $x=3,4$ .

Factorization method:

  $x^2-7x+12=0$

To solve the equation by factorization, split the middle term of the equation.

  $\begin{array}{c}{x}^2-7x+12=0\\ x^2-3x-4x+12=0\\ x\left(x-3\right)-4\left(x-3\right)=0\\ \left(x-3\right)\left(x-4\right)=0\end{array}$

Solve the factors for x .

  $\begin{array}{c}x-3=0\\ x=3\end{array}$

And, $\begin{array}{c}x-4=0\\ x=4\end{array}$

Thus, the solutions of the equation are $x=3,4$ .

The factorization method of solving the equation is preferable.

Hence, the solutions are $x=3,4$ and the factorization method is preferable.