Question

How do you multiply \[\left( {7x - 6} \right)\left( {5x + 6} \right)\]?

Answer 1

Hint: In order obtain the multiplied form of the above binomial expression, use the order of expanding as FOIL i.e. First,Outside,Inside,Last and later combine all the like terms and rearrange the equation in the standard quadratic equation of the form $a{x^2} + bx + c$ to get the required answer.

Complete step-by-step solution:
We are given a polynomial having one variable $x$ in the term.
Let’s suppose the function given be $f\left( y \right)$
$f\left( x \right) = \left( {7x - 6} \right)\left( {5x + 6} \right)$
First we have to expand the above equation, to expand any binomial expression there is an order by which we can expand the expression easily. You can use the acronym FOIL which means First, Outside,Inside,Last to remember the order of expanding .
\[
\Rightarrow f\left( x \right) = \left( {7x - 6} \right)\left( {5x + 6} \right) \\
 \Rightarrow 7x\left( {5x + 6} \right) - 6\left( {5x + 6} \right) \\
 \Rightarrow 35{x^2} + 42x - 30x - 36 \\
 \]
Now Simplification means to combine all the like terms in the equation.
So combining all the terms, and rearranging the above quadratic equation in the form of standard quadratic equation as $a{x^2} + bx + c$
Our equation becomes
\[f\left( x \right) = 35{x^2} + 12x - 36\]

Therefore, the multiplied form of the given equation is \[f\left( x \right) = 35{x^2} + 12x - 36\]

Additional Information:
Quadratic Equation: A quadratic equation is an equation which can be represented in the form of $a{x^2} + bx + c$ where $x$ is the unknown variable and a,b,c are the numbers known where $a \ne 0$. If $a = 0$then the equation will become a linear equation and will no longer be quadratic .
The degree of the quadratic equation is of the order 2.
In order to determine the roots to a quadratic equation, there are couple of ways,
1.Using splitting up the middle term method:
let $a{x^2} + bx + c$
 calculate the product of coefficient of ${x^2}$ and the constant term and factorise it into two factors in a way that either addition or subtraction of the two gives the middle term and multiplication gives the product value.
2.You can also alternatively use a direct method which uses Quadratic Formula to find both roots of a quadratic equation as
$x1 = \dfrac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}$ and $x2 = \dfrac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}$
$x_1,x_2$ are root to quadratic equation $a{x^2} + bx + c$

Note:
1. Don’t Forgot cross-check your answer.
2. Like terms are the terms having the same variable and power.
3.Like terms may have coefficients different.
4. Make sure all the terms are combined properly.