A quadratic equation that can be solved by factoring is [tex]x^2 + 5x + 6[/tex]. If factors are not easily detected, then we can use the ultimate formula for solving any quadratic equation.
How to find the solution to a standard quadratic equation?
Suppose the given quadratic equation is
[tex]ax^2 + bx + c = 0[/tex]
Then its solutions are given as
[tex]x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
How to find the factors of a quadratic expression?
If the given quadratic expression is of the form [tex]ax^2 + bx + c = 0[/tex],
then its factored form is obtained by two numbers alpha( α ) and beta( β) such that:
[tex]b = \alpha + \beta \\ ac =\alpha \times \beta[/tex]
Then writing b in terms of alpha and beta would help us getting common factors out.
Its not always possible to find such real numbered factors because they might not exist at all, example for [tex]x^2 = -1[/tex], no factors can be made by real numbers.
Since we need to find such quadratic equation which can be solved by factoring, we can just construct it backwards from two linear factors.
Thus, let we take two factors as (x+3) and (x+2), then:
[tex](x+3)(x+2) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6[/tex]
Thus, a quadratic equation that can be solved by factoring is [tex]x^2 + 5x + 6[/tex]
If factors are not easily detected, then we can use the ultimate formula for solving any quadratic equation.
Learn more about quadratic equations here:
https://brainly.com/question/3358603
