Question

How do you factor and solve \[{x^2} - 10x = - 16\] ?

Answer 1

Hint: The above given equation is a quadratic equation that is a polynomial of degree two. So, the equation has at most two solutions or roots. There are mainly three different methods to solve the equation – factoring, completing the square and using the quadratic formula $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ .

Complete step by step solution:
The general form of a quadratic equation is $ a{x^2} + bx + c = 0 $ . So, we first proceed by rearranging the terms of the given equation to get the standard form.
 \[{x^2} - 10x = - 16\]
 \[ \Rightarrow {x^2} - 10x + 16 = 0\] -- (1)
Next, we will compare this equation with the standard form and find the values of $ a $ , $ b $ and $ c $ .
So, we get, $ a = 1,{\text{ }}b = - 10{\text{ and }}c = 16 $ .
Next, we have to find two numbers whose product gives $ ac $ and sum equals $ b $ . Here $ ac = 1 \times 16 = 16 $ . So, the possible pairs of numbers whose product is 16 are,
$ 1 \times 16,{\text{ }} - 1 \times ( - 16),{\text{ }}2 \times 8,{\text{ }} - 2 \times ( - 8),{\text{ }}4 \times 4{\text{ and - 4}} \times {\text{( - 4)}} $ .
We can observe that, among these only the sum of the pair $ - 2{\text{ and }} - 8 $ equals $ - 10 $ . So, we will split the $ - 10x $ term into $ - 2x $ and $ - 8x $ .
Then, from equation (1) we have, \[{x^2} - 2x - 8x + 16 = 0\]
 \[ \Rightarrow x(x - 2) - 8(x - 2) = 0\]
 \[ \Rightarrow (x - 2)(x - 8) = 0\] (by taking the common factors outside)
So, the factors of the equation are \[x - 2{\text{ and }}x - 8\] .
We will now equate the factors to zero and find the solution of the quadratic equation.
 \[x - 2 = 0\] and \[x - 8 = 0\]
 \[ \Rightarrow x = 2\] and \[x = 8\] .
Hence the solution of the given quadratic equation is \[x = 2\] and \[x = 8\] .
So, the correct answer is “ \[x = 2\] and \[x = 8\] ”.

Note: At times we may find it difficult to find those numbers whose product equals $ac$ and sum equals $b$. Factoring becomes tedious in those cases. An easier way to solve would be to use the quadratic formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. The method is illustrated below for our question.
Here, we put the values of $a$, $b$and $c$in the formula to get,
$x = \dfrac{{ - ( - 10) \pm \sqrt {{{( - 10)}^2} - 4(1)(16)} }}{{2(1)}}$
$ \Rightarrow x = \dfrac{{10 \pm \sqrt {100 - 64} }}{2}$
$ \Rightarrow x = \dfrac{{10 \pm \sqrt {36} }}{2}$
$ \Rightarrow x = \dfrac{{10 \pm 6}}{2}$
$ \Rightarrow x = \dfrac{{10 + 6}}{2}{\text{ and }}x = \dfrac{{10 - 6}}{2}$
\[ \Rightarrow x = \dfrac{{16}}{2}{\text{ and }}x = \dfrac{4}{2}\]
\[ \Rightarrow x = 8{\text{ and }}x = 2\]
This is the same as the solution obtained by factoring the equation.