Question

Find the factors of \[2{x^2} + 7x - 4\].

Answer 1

Hint: Given an equation is a quadratic equation therefore this equation will be having two roots. A quadratic is a polynomial that looks like \[a{x^2} + bx + c\]where “a”, “b” and “c” are just numbers. We will need to find two numbers that will multiply to be equal to the constant term “c”, and will also add up to equal “b”, the coefficient on the linear \[x\]term in the middle.

Complete step by step answer:
We need to solve for the values of the variable that make the equation true.
To solve quadratics by factoring, we use something called “the zero-product property”. The property says if we multiply two or more things together and the result is equal to zero then at least one of those things that we multiplied must also have been equal to zero.
Finding a pair of “a” and “c” we have \[2 \times 4 = 8\]which differ by\[b = 7\].
The pair \[8,1\]works. Using this pair to split the middle term and factor by grouping we have,
\[
  2{x^2} + 7x - 4 \\
   \Rightarrow 2{x^2} + 8x - x - 4 \\
   \Rightarrow \left( {2{x^2} + 8x} \right) - \left( {x + 4} \right) \\
   \Rightarrow 2x\left( {x + 4} \right) - 1\left( {x + 4} \right) \\
   \Rightarrow \left( {2x - 1} \right)\left( {x + 4} \right) \\
 \]
Hence, the factors of \[2{x^2} + 7x - 4\]are \[\left( {2x - 1} \right)\]and\[\left( {x + 4} \right)\].

Note:
 One of those factors must have been equal to zero if the product itself equals to zero so that we can set the factors equal to zero. If the product of factors equal to anything non-zero, then we cannot make any claim about the values of the factors. Therefore, when solving quadratic equations by factoring, we must always have the equation in the form of a quadratic equation equals to zero before we make any attempt to solve the quadratic equation by factoring.