Question

Solve: ${x^2} + 3x - 4 = 0$?

Answer 1

Hint: The given problem requires us to solve an equation. There are various methods that can be employed to solve a quadratic equation like completing the square method, using quadratic formulas and by splitting the middle term. The given equation can be solved easily using splitting the middle term method.

Complete step by step solution:
In the given question, we are required to solve the equation ${x^2} + 3x - 4 = 0$.
Consider the equation ${x^2} + 3x - 4 = 0$.
The equation can be solved by various methods such as completing the square method, splitting the middle term and using the quadratic formula. Solving the quadratic equation by splitting the middle term, we have to split the middle term into two terms such that the sum of the terms gives us the original middle term and product of the terms gives us the product of the constant term and coefficient of ${x^2}$.
So, ${x^2} + 3x - 4 = 0$
$ \Rightarrow {x^2} + \left( {4 - 1} \right)x - 4 = 0$
$ \Rightarrow {x^2} + 4x - x - 4 = 0$
We split the middle term $ - 4x$ into two terms $ - 3x$ and $ - x$ since the product of these terms, $3{x^2}$ is equal to the product of the constant term and coefficient of ${x^2}$ and sum of these terms gives us the original middle term, $ - 4x$.
$ \Rightarrow x\left( {x + 4} \right) - \left( {x + 4} \right) = 0$
$ \Rightarrow \left( {x - 1} \right)\left( {x + 4} \right) = 0$
Now, either $\left( {x + 4} \right) = 0$ or $\left( {x - 1} \right) = 0$.
So, either $x = - 4$ and $x = 1$.
Hence, the roots of the equation ${x^2} + 3x - 4 = 0$ are $x = \left( { - 4} \right)$ and $x = 1$.

Note: Quadratic equations are the polynomial equations with degree of the variable or unknown as $2$. Quadratic equations can be solved by splitting the middle term, using the quadratic formula and completing the square method. Some equations don’t appear to be quadratic but can be reduced into quadratic equations using substitution.