Chapter A.6, Problem 144AYU

To determine

To find: Three ways you might solve a quadratic equation. State your preferred method; explain why you chose it.

Answer to Problem 144AYU

There are three methods you can use to solve a quadratic equation:

Factoring

The factors of a quadratic equation will be $\left(qx+r\right)$ and $\left(sx+t\right)$. If the solutions are all integers, you may be able to quickly find $q$, $r$, $s$ and $t$. The advantage of this method is that factoring can be very fast. The disadvantage is that factoring may not work; for instance, factoring will not find solutions that are not integers.

Completing the square

Completing the square is a multistep process. The main idea is to convert the original equation into one of the form ${\left(x + a\right)}^2 = b$, where $a$ and $b$ are constants. The advantage of this method are that it always works and that completing the square gives some insight into how algebra works more generally. The disadvantage is that this method is complex.

Quadratic Formula

The quadratic formula is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. The advantages of this method are that the quadratic formula always works and is straightforward. The disadvantages are that the formula provides no insight and can become a rote technique.

I choose Quadratic formula, since in this method we can solve any quadratic equation by using formula. It gives either real or complex solution.

Explanation of Solution

There are three methods you can use to solve a quadratic equation:

Factoring

The factors of a quadratic equation will be $\left(qx+r\right)$ and $\left(sx+t\right)$. If the solutions are all integers, you may be able to quickly find $q$, $r$, $s$ and $t$. The advantage of this method is that factoring can be very fast. The disadvantage is that factoring may not work; for instance, factoring will not find solutions that are not integers.

Completing the square

Completing the square is a multistep process. The main idea is to convert the original equation into one of the form ${\left(x + a\right)}^2 = b$, where $a$ and $b$ are constants. The advantage of this method are that it always works and that completing the square gives some insight into how algebra works more generally. The disadvantage is that this method is complex.

Quadratic Formula

The quadratic formula is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. The advantages of this method are that the quadratic formula always works and is straightforward. The disadvantages are that the formula provides no insight and can become a rote technique.

I choose Quadratic formula, since in this method we can solve any quadratic equation by using formula. It gives either real or complex solution.