Question

How do you simplify \[(x - 5)(x + 3)?\]

Answer 1

Hint: According to the question we have to determine the values of x or in other words we can say that we have to simplify the given expression which is \[(x - 5)(x + 3)\]. So, first of all we have to open the smaller brackets of the terms to multiply them all.
Now, we have to multiply $x$ with $(x + 3)$ and the same as we have to multiply -5 with $(x + 3)$.
Now, after the multiplication we have to add the terms which can be added and subtract the terms which can be subtracted.
Now, we will obtain a quadratic expression and to solve the quadratic expression we have to use the formula to find the roots of the quadratic expression which is as mentioned below:

Formula used: $ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$……………….(A)
Where, a is the coefficient of ${x^2}$, a is the coefficient of $x$ and c is the constant.

Complete step-by-step solution:
Step 1: First of all we have to open the smaller brackets of the terms to multiply them all as mentioned in the solution hint. Hence,
\[ \Rightarrow x(x + 3) - 5(x + 3)...............(1)\]
Step 2: Now, we have to multiply each of the terms of the expression (1) which is as obtained in the solution step 1. Hence,
$
   \Rightarrow {x^2} + 3x - 5x - 15 \\
   \Rightarrow {x^2} - 2x - 15................(2)
 $
Step 3: Now, to solve the quadratic expression we have to use the formula (A) to find the roots of the quadratic expression which is as mentioned in the solution hint. Hence, on substituting all the values in the formula (A),
$ \Rightarrow x = \dfrac{{ - ( - 2) \pm \sqrt {{{( - 2)}^2} - 4 \times 1 \times ( - 15)} }}{{2 \times 1}}$
On solving the expression as obtained just above,
$
   \Rightarrow x = \dfrac{{2 \pm \sqrt {4 - 60} }}{2} \\
   \Rightarrow x = \dfrac{{2 \pm \sqrt { - 56} }}{2}
 $
Now, we have to take 2 as a common term form the expression as obtained just above,
$
   \Rightarrow x = \dfrac{{2 \pm 2\sqrt { - 14} }}{2} \\
   \Rightarrow x = \dfrac{{1 \pm \sqrt { - 14} }}{2}
 $

Hence, with the help of the formula (A) we have determined the roots of the expression or we have simplified the expression \[(x - 5)(x + 3)\] and the solution is $x = \dfrac{{1 \pm \sqrt { - 14} }}{2}$.

Note: On solving the quadratic expression only two roots or zeros can be formed and these both of the obtained roots or zeroes will satisfy the expression as given in the question.
We can also obtain the roots by finding the L.C.M of the constant term and the coefficient of ${x^2}$ and then we have to make the factors obtained to the coefficient of x.