Chapter 8.4, Problem 1E

To determine

To describe : two methods for solving a quadratic equation. Mention the advantage and disadvantage of using each method.

Answer to Problem 1E

Final answer

Explanation of Solution

Given information :

Two methods for solving a quadratic equation and their advantages and disadvantages.

The two methods to solve a quadratic equation are:

1. Completing the square:

To solve a quadratic equation of the form $a{x}^2+bx+c=0\text{ (}a\ne 0)$ , transform the equation into the form ${(x+q)}^2=r$ .

Step 1. Transform the equation so that the constant term c is alone on the right side.

Step 2. If a, the coefficient of the second-degree term, is not equal to $1$ , then divide both sides by a .

Step 3. Add the square of half the coefficient of the first-degree term, $\left(\frac{b^2}{2a}\right)$ , to both sides. (Completing the square)

Step 4. Factor the left side as the square of a binomial.

Step 5. Complete the solution using the fact that ${(x+q)}^2=r$ is equivalent to $x+q=\pm \sqrt{r}$

Advantages of this method: The method always works.

Disadvantage of this method: The method is lengthy , sometimes complex and takes time.

2. Quadratic formula
To solve a quadratic equation of the form $a{x}^2+bx+c=0\text{ (}a\ne 0)$ , the following formula can be used:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Advantage of this method: The method always works giving accurate answers. The formula finds the roots of the quadratic equations.

Disadvantage of this method: The calculation may be difficult without a calculator.