Question

How do you factor \[10{{x}^{2}}+35x-4x-14\] by grouping?

Answer 1

Hint: We already know that a function can only be factorized if it has real roots. So, before using the grouping method to factorize the function, we will check if it has real roots or not. The degree of the given equation is 2, it means that the equation is a quadratic. We know that for a quadratic equation to have real roots, it should satisfy the condition \[{{b}^{2}}-4ac\ge 0\]. We will check if the equation satisfies this condition or not.

Complete step by step answer:
We are given the equation \[10{{x}^{2}}+35x-4x-14\]. We will first check if the equation has real roots or not. The degree of the given equation is 2, it means that the equation is a quadratic. We know that for a quadratic equation to have real roots, it should satisfy the condition \[{{b}^{2}}-4ac\ge 0\].
Simplifying the given equation, we can write it as \[10{{x}^{2}}+31x-14\].
For this equation, we have \[{{\left( 31 \right)}^{2}}-4(10)(-14)=961+560=1521\]. This value is non-negative. Hence, the equation has real roots. Now, we can factorize the equation using grouping method as
\[10{{x}^{2}}+35x-4x-14\]
Taking \[5x\]and \[-2\] in common from the first two terms, and last two terms respectively, we can simplify the expression as
\[\Rightarrow 5x\left( 2x+7 \right)-2\left( 2x+7 \right)\]
Taking the term \[\left( 2x+7 \right)\] in common from the whole expression, we get
\[\Rightarrow \left( 5x-2 \right)\left( 2x+7 \right)\]
This is the factored form of the given expression.

Note: For this problem, we are already given the terms such that we can take their HCF in common. For general questions, we have to split the middle term of a quadratic equation such that we can factorize it using a grouping method.