Question

Simplify:
$(x - 3)(2x + 1) = x(x + 5)$

Answer 1

Hint:By simplifying the equation, we mean multiplying the terms within the brackets with each other that is each term in one bracket is multiplied with each term in the other bracket then we place the like terms with each other and apply the mathematical operations like addition and subtraction, present between them. This way we can solve the given question.

Complete step by step answer:
We are given that $(x - 3)(2x + 1) = x(x + 5)$

For multiplying the terms within the brackets with each other, on the left-hand side, we first multiply x and -3 one by one with the other bracket and on the right-hand side we multiply x with the terms in the bracket, we get –

$
x(2x + 1) - 3(2x + 1) = {x^2} + 5x \\
2{x^2} + x - 6x - 3 = {x^2} + 5x \\
$

Now, we rearrange the obtained terms and place like terms together –
$
\Rightarrow 2{x^2} - {x^2} - 5x - 5x - 3 = 0 \\
\Rightarrow {x^2} - 10x - 3 = 0 \\
$

Hence, the simplified form of $(x - 3)(2x + 1) = x(x + 5)$ is ${x^2} - 10x - 3 = 0$

Note:The equation obtained above is a quadratic equation and can be written as $f(x) = {x^2} - 10x - 3$ . .

So, after simplifying the obtained equation we can find out its roots. The points on the x-axis at which the y-coordinate of the function is zero are called the roots of the equation or we can also say that the roots of an equation are the points on the x-axis that is the roots are simply the x- intercepts. In a polynomial equation, the highest exponent of the polynomial is called its degree. And according to the Fundamental Theorem of Algebra, a polynomial equation has exactly as many roots as its degree. So, two roots of the above equation exist.

The roots of an equation can be found out by factoring the equation and also by a special formula called completing the square method.