Solve the Quadratic Equation: x²-10x=-16 Step-by-Step

Solve the given equation:

$x^2-10x=-16$

First, let's arrange the equation by moving terms:

$x^2-10x=-16 \\ x^2-10x+16 =0$

Note that the coefficient of the squared term is 1, therefore, we can (try to) factor the expression on the left side by using quick trinomial factoring:

Let's look for a pair of numbers whose product equals the free term in the expression, and whose sum equals the coefficient of the first-degree term, meaning two numbers $m,\hspace{2pt}n$ that satisfy:

$m\cdot n=16\\ m+n=-10\\$From the first requirement, namely - the multiplication, we observe that the product of the numbers we're looking for must yield a positive result, therefore we can conclude that both numbers must have the same sign, according to multiplication rules. Remember that the possible factors of 16 are the number pairs 4 and 4, 2 and 8, or 16 and 1. Meeting the second requirement, along with the fact that the signs of the numbers we're looking for are identical leads us to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases} m=-8\\ n=-2 \end{cases}$

Therefore we'll factor the expression on the left side of the equation to:

$x^2-10x+16 =0 \\ \downarrow\\ (x-8)(x-2)=0$

From remember that the product of expressions will yield 0 only if at least one of the multiplied expressions equals zero,

Therefore we'll obtain two simple equations and solve them by isolating the unknown in each:

$x-8=0\\ \boxed{x=8}$

or:

$x-2=0\\ \boxed{x=2}$

Let's summarize the solution of the equation:

$x^2-10x=-16 \\ x^2-10x+16 =0 \\ \downarrow\\ (x-8)(x-2)=0 \\ \downarrow\\ x-8=0\rightarrow\boxed{x=8}\\ x-2=0\rightarrow\boxed{x=2}\\ \downarrow\\ \boxed{x=8,2}$

Therefore the correct answer is answer B.