Question

How do you solve $\left| {5x - 5} \right| = 15?$

Answer 1

First of all we will take square on both the sides of the equation. Here we will use the concept of splitting the middle terms and first making the pair of two terms and then finding the common factors from the paired terms and finally the factors for the given expression.

Complete step-by-step solution:
Take the given expression:
$\left| {5x - 5} \right| = 15$
Take square on both sides of the equation.
$\Rightarrow {\left| {(5x - 5)} \right|^2} = {15^2}$
Apply the whole square in the above equation. When the number is multiplied with the same number twice it is known as square.
$\Rightarrow (5x - 5)(5x - 5) = {15^2}$
Take common factors from the left hand side of the equation and apply the square of the number on the right hand side of the equation.
$\Rightarrow 25(x - 1)(x - 1) = 225$
Simplify the above equation:
$\Rightarrow 25({x^2} - 2x + 1) = 225$
Multiply the constant inside the bracket.
$\Rightarrow 25{x^2} - 50x + 25 = 225$
Move the term from the right hand side of the equation to the left hand side of the equation. When any term is moved from one side to the opposite side, then the sign of the term is also changed. Positive term changes to the negative term and vice-versa.
$\Rightarrow 25{x^2} - 50x + 25 - 225 = 0$
Make the pair of like terms in the above equation.
$\Rightarrow 25{x^2} - 50x + \underline {25 - 225} = 0$
While simplifying between the like terms, when there is a positive term and the negative term you have to do subtraction and the sign of a bigger digit.
$\Rightarrow 25{x^2} - 50x - 200 = 0$
Take common multiples from all the terms in the above equation.
$\Rightarrow {x^2} - 2x - 8 = 0$
The above equation can be re-written as –
$\Rightarrow {x^2} - \underline {4x + 2x} - 8 = 0$
Again, making the pair of two first terms and last two terms.
$\Rightarrow \underline {{x^2} - 4x} + \underline {2x - 8} = 0$
Take common multiple from the above two paired terms.
$\Rightarrow x(x - 4) + 2(x - 4) = 0$
Simplify the above equation:
$\Rightarrow (x - 4)(x + 2) = 0$
Simplify the above equation:
$\Rightarrow x = 4$or $\Rightarrow x = ( - 2)$
This is the required solution.

Note: Here we were able to split the middle term and find the factors but in case it is not possible then we can find factors by using the formula$x = \frac{{ - b \pm \sqrt \Delta }}{{2a}}$ and considering the general form of the quadratic equation $a{x^2} + bx + c = 0$ . Be careful about the sign convention and simplification of the terms in the equation.