Question

How do you solve the equation $(x - 10)(x + 5) = 0?$

Answer 1

Hint: First of all we will open the brackets and will simplify it then framing the equation 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 equation: $(x - 10)(x + 5) = 0$
Open the brackets
$ \Rightarrow x(x + 5) - 10(x + 5) = 0$
When you open the bracket and there is a positive sign outside the bracket then there will be no change in the sign of the terms inside the bracket but when there is a negative sign outside the bracket then the sign of all the terms inside the bracket changes. Positive terms become negative and vice-versa.
$ \Rightarrow {x^2} + 5x - 10x - 50 = 0$
Simplify the equation among the like terms.
$ \Rightarrow {x^2} + \underline {5x - 10x} - 50 = 0$
When you simplify between one positive and one negative term then you have to do subtraction and give signs of bigger numbers.
$ \Rightarrow {x^2} - 5x - 50 = 0$
Again, to solve and find the factors of the above equation will split the middle term.
To split the middle term, we will split in such a way that the product of the two split terms will be equal to the product of first and the last terms. Also, with the same sign convention.
When you have to simplify one negative term and one positive term then for the resultant value you have to perform subtraction and sign of a bigger digit. In multiplication if there is one negative and one positive term then the resultant product will be with negative sign.
So,
$
\Rightarrow + 5 = 10 - 5 \\
\Rightarrow - 50 = (10)( - 5) \\
 $
 Take the simplified expression,
$ \Rightarrow {x^2} - 5x - 50 = 0$
The above expression can be re-written by splitting the middle term,
$ \Rightarrow {x^2} + \underline {10x - 5x} - 50 = 0$
Make pair of first two and last two terms,
$ \Rightarrow \underline {{x^2} + 10x} - \underline {5x - 50} = 0$
Take out common factors from the above two paired terms.
$x(x + 10) - 5(x + 10) = 0$
Take out common from the above equation, Take common multiple common from the first bracket from both the terms.
$(x + 10)(x - 5) = 0$
$
   \Rightarrow x + 10 = 0 \\
   \Rightarrow x = ( - 10) \\
 $
Or
$
   \Rightarrow x - 5 = 0 \\
   \Rightarrow x = 5 \\
 $
Hence, the above solution implies, $x = ( - 10){\text{ or x = 5}}$

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\[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\] 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.