Question

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

Answer 1

Hint: In this question, they have given us an equation and asked us to factor and solve it. First, we need to alter the given equation by transferring all the terms to one side. This will give us a standard equation. Then, we will find the required factors and we will alter the equation in such a way that we can find the commons among the terms to factor and solve it. After taking the commons out we will be able to find the values easily.

Complete step-by-step solution:
The given equation is ${x^2} - 10x = - 16$ and we need to factor this and solve the equation.
First, we should alter the given equation by transferring $ - 16$ to the other side.
${x^2} - 10x = - 16$
$\Rightarrow$${x^2} - 10x + 16 = 0$
Now, we need to split the middle term in such a way that the two terms multiply to give us $16$ and add or subtract to give us $ - 10$ .
Splitting $ - 10x$ into $ - 8x$ and $ - 2x$ can be such factors and they can help us find the common terms in the equation.
$\Rightarrow$${x^2} - 8x - 2x + 16 = 0$
As we can see that $x$ and $ - 2$ is common, we will take it out respectively.
$\Rightarrow$$x\left( {x - 8} \right) - 2\left( {x - 8} \right) = 0$
Now, $x - 8$ is common in the terms, taking it out we get
$\Rightarrow$$(x - 8)(x - 2) = 0$
Solving this, we get
$\Rightarrow$$x = 8,x = 2$

Therefore, the roots are $x = 8,x = 2$.

Note: Alternative method:
We can also solve this problem by using the quadratic equation formula.
Quadratic equation formula:
 ${\text{x = }}\dfrac{{{{ - b \pm }}\sqrt {{{\text{b}}^{\text{2}}}{\text{ - 4ac}}} }}{{{\text{2a}}}}$
Rewriting the given equation, we get, ${x^2} - 10x + 16 = 0$
Clearly in ${x^2} - 10x + 16 = 0$ ,
${\text{a = 1}}$,
${\text{b = - 10}}$,
${\text{c = 16}}$.
Now applying ${x^2} - 10x + 16 = 0$ in $\dfrac{{{{ - b \pm }}\sqrt {{{\text{b}}^{\text{2}}}{\text{ - 4ac}}} }}{{{\text{2a}}}}$ ,
  ${\text{x = }}\dfrac{{ - ( - 10) \pm \sqrt {{{( - 10)}^2} - 4(1)(16)} }}{{2(1)}}$
Simplifying the numerator, we get,
${\text{x = }}\dfrac{{10 \pm \sqrt {100 - 64} }}{2}$
${\text{x = }}\dfrac{{10 \pm \sqrt {36} }}{2}$
Taking the square root in the numerator, we get
$x = {\text{ }}\dfrac{{10 \pm 6}}{2}$
Now we have to expand the expression into two, as there is a $ \pm $ in the expression. One becomes plus and the other becomes minus.
 $x = {\text{ }}\dfrac{{10 + 6}}{2},{\text{ }}\dfrac{{10 - 6}}{2}$
Solving this we get,
$x = 8,2$
Therefore, the roots are $x = 8,x = 2$.