Solve the Quadratic Equation: x²-10x-4=10-5x Step-by-Step

Let's solve the given equation:

$x^2-10x-4=10-5x$

First, let's rearrange the equation by combining like terms:

$x^2-10x-4=10-5x \\ x^2-10x-4-10+5x=0 \\ x^2-5x-14=0$

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

$m\cdot n=-14\\ m+n=-5\\$From the first requirement mentioned, that is - from the multiplication, we notice that the product of the numbers we're looking for needs to yield a negative result, therefore we can conclude that the two numbers have different signs according to multiplication rules. Furthermore the possible factors of 14 are 7 and 2 or 14 and 1. Fulfilling the second requirement mentioned, along with the fact that the numbers we're looking for have different signs leads to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases}m=-7\\ n=2\end{cases}$

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

$x^2-5x-14=0 \\ \downarrow\\ (x-7)(x+2)=0$

Remember that the product of expressions equals 0 only if at least one of the multiplied expressions equals zero,

Therefore we obtain two simple equations and solve them by isolating the unknown term:

$x-7=0\\ \boxed{x=7}$

or:

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

Let's summarize the solution of the equation:

$x^2-10x-4=10-5x \\ x^2-5x-14=0 \\ \downarrow\\ (x-7)(x+2)=0 \\ \downarrow\\ x-7=0\rightarrow\boxed{x=7}\\ x+2=0\rightarrow\boxed{x=-2}\\ \downarrow\\ \boxed{x=7,-2}$

Therefore the correct answer is answer A.