Question

How do you know when to use factoring or quadratic formula?

Answer 1

Hint: When we are given a quadratic equation to solve, we can either solve it by factoring method or by the quadratic formula. Since, we know that all the equations can be solved by the quadratic equation but it is time consuming. So, think about the conditions where factoring would be easier and less time consuming and the conditions where using the quadratic formula would be more apt

Complete step-by-step answer:
(i)
When we are given a quadratic equation and we have to find its roots, sometimes it is mentioned in the question which method of solution to use when solving the given equation. The other times, when it is not indicated in the question, the choice is on us to choose the correct method in order to calculate its roots.
Now, as we know that the when we have our quadratic equation in the form of
 $a{x}^2+bx+c=0$ , its roots will be $x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$ which is called as the quadratic formula. As this formula is applicable in every condition, we can always apply this formula. Especially, when we have complicated coefficients like the coefficients which involve roots or when the factoring does not seem possible or if we cannot easily find the factors of the equation. This usually happens when the coefficients are big numbers or when the quadratic equation has imaginary roots.
But since, using the quadratic formula is much time consuming and we have to more careful with arithmetic i.e., calculations while using the formula, opting factoring method in the correct situation can save us some time as factoring is usually faster and less prone to arithmetic mistakes.
Therefore, if the coefficient of $x^2$ and the constant element have relatively few factors, time invested in attempting to factor the quadratic is usually worthwhile. It is also useful when the constant term $c$ is missing. In this case, we can easily factorize the given quadratic equation.
Therefore, when we come across a quadratic equation of the standard form $a{x}^2+bx+c=0$ , we should first attempt the factoring method. But we should not waste much time in finding the factors, so if it takes more than $60$ seconds to find a factor, we should move to another method which is using the quadratic formula as it is applicable on all kinds of quadratic equations.

Note: We can also use the discriminant $D=b^2-4ac$ , to check the nature of the roots. If $D<0$ , the quadratic equation has imaginary roots, so don’t waste your time in factoring. Instead, directly apply the quadratic formula to obtain the imaginary roots. While if $$D>0$$, the quadratic equation has two different roots, so if the coefficients are small, we can attempt to find the factors. And if we have $D=0$ , the quadratic equation has two equal roots. In this situation either we can directly calculate the root by applying the formula $x={\displaystyle \frac{-b}{2a}}$ or by converting the equation $a{x}^2+bx+c=0$ into a perfect square.