Question

How do you factor the expression \[3{x^2} + 7x - 6\]?

Answer 1

Hint: In this question, we have to factor the given expression.
First we need to know that the given expression is a quadratic expression so we can solve it by splitting the middle term. First we need to multiply the coefficient of the first term and the constant term then need to factor that term whose sum equals the coefficient of the middle term. Then factoring the term we will get the required solution.

Complete step by step answer:
The given expression is \[3{x^2} + 7x - 6\].
We need to factor the given expression.
Step I:
The first term in the expression is \[3{x^2}\].
The second term in the expression is \[7x\].
The third term in the expression is \[ - 6\].
Step II:
Multiply the coefficient of the first term by the constant term we get,
\[3 \times \left( { - 6} \right) = - 18\]
Step III:
Find two factors of \[ - 18\] whose sum equals the coefficient of the middle term, which is\[7\].
\[7 = 9 - 2\]
Step IV:
Rewrite the polynomial by splitting the middle term using the two factors\[ - 9, - 2\], we get,
\[3{x^2} + 9x - 2x - 6\]
Step V:
Write the expression in terms of factors
\[3x\left( {x + 3} \right) - 2\left( {x + 3} \right)\]
Taking common from first two terms and last two terms
Or,\[\left( {x + 3} \right)\left( {3x - 2} \right)\]

Hence, we get the factor the given expression \[3{x^2} + 7x - 6\] is \[\left( {x + 3} \right)\left( {3x - 2} \right)\].

Note: Quadratic equation: In algebra a quadratic equation is any equation that can be rearranged in standard form as
\[a{x^2} + bx + c = 0\], where x represents an unknown and a, b, c represent known numbers where a ≠ 0.
If \[a{\text{ }} = {\text{ }}0\] then it will become a linear equation not quadratic as there is no \[a{x^2}\]term.
Quadratic equations can be solved by a middle term process.
In this method we will Split the middle term in such a way that either it will be represented as addition of two numbers or subtraction of two numbers, then making the quadratic equation as a multiplication of two factors and solving the two factors we will get the solution.