Question

Factorize the given equation $6{x^2} + 5x - 6$

Answer 1

Hint: $6{x^2} + 5x - 6$ is a quadratic polynomial because it has degree 2. A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
In general every polynomial of the form $a{x^2} + bx + c$ is a quadratic polynomial. We solve this quadratic polynomial by factorization method.

Complete step-by-step answer:
To factorize this expression $p(x) = 6{x^2} + 5x - 6$ here we follow the following method:
The coefficient of x is 5 and the product of the coefficient of ${x^2}$ and the constant term is 36.
So, we need two numbers which on addition or subtraction gives 5 and on multiplication 36.
These numbers are 9 and 4.beacuse 9-4=5 and 9× 4=36.
So, $p(x) = 6{x^2} + (9 - 4)x - 6$
$ = 6{x^2} + 9x - 4x - 6$
$p(x) = 6{x^2} + 5x - 6................(i)$
Now, the factor of $6{x^2} = 2 \times 3 \times x \times x$
Factors of $9x = 3 \times 3 \times x$
Factor of $ - 4x = - 2 \times 2 \times x$
Factor of -6 = -2× 3
Now in (i) from the first two terms 3x is common so we can take 3x common from the first two terms.
In the last two terms, -2 is common. So we can take -2 common from the last two terms.
∴ $p(x) = 3x(2x + 3) - 2(2x + 3)$
Now (2x+3) is common in both terms.
∴ $p(x) = (2x + 3)(3x - 2)$
So, a factor of $6{x^2} + 5x - 6$ are $(2x + 3)(3x - 2)$

Note: An algebraic expression of type $a{x^2} + bx + c$,$a \ne 0$ is known as quadratic polynomial, or we can say that a polynomial with degree 2 is called a quadratic polynomial. Remember that a polynomial is an algebraic expression that consists of terms in the form axnaxn.