- Factoring a quadratic equation can be defined as the process of breaking the equation into the product of its factors.
[tex] \:❏ \: \: \LARGE{\rm{{{\color{orange}{6 {x }^{2} \: + \: 7x \: - 3}}}}}[/tex]
- Factorize the equation by breaking down the middle term.
[tex] \large \blue\implies \tt \large \: 6 {x }^{2} \: + \: 7x \: - 3[/tex]
- Let’s identify two factors such that their sum is 7 and the product is -18.
Sum of two factors = 7 = 9 - 2
Product of these two factors = 9 × (-2) = 18
- Now, split the middle term.
[tex]\large \blue\implies \tt \large \:6 {x}^{2} \: + \: 9x \: - \: 2x \: - \: 3[/tex]
- Take the common terms and simplify.
[tex] \: \\ \large\blue\implies \tt \large \:3x(2x \: + \: 3)\: -1(2x \: + \: 3)[/tex]
[tex] \\ \large\blue\implies \tt \large \:(3x \: - \:1 ) \quad \: (2x \: + \: 3) \: = \: 0[/tex]
Thus, (3x - 1) and (2x + 3) are the factors of the given quadratic equation.
- Solving these two linear factors, we get
[tex]\large\blue\implies \tt \large \:x \: = \: \frac{1}{3} \: \: , \: \: \frac{ - 3}{2} \\ [/tex]
