Question

Solve the equation, if possible. (If all real numbers are solutions, enter REALS. If there is no solution, enter NO SOLUTION.)

Answer 1

Answer:

The equation $x\left(x\:+\:6\right)\:=\:\left(x\:+\:3\right)^2$ has NO SOLUTION, and is called contradiction.

Step-by-step explanation:

Given the equation

$x\left(x\:+\:6\right)\:=\:\left(x\:+\:3\right)^2$

$\mathrm{Expand\:}x\left(x+6\right):\quad x^2+6x$

$\mathrm{Expand\:}\left(x+3\right)^2:\quad x^2+6x+9$

so

$x^2+6x=x^2+6x+9$

$\mathrm{Subtract\:}x^2+6x\mathrm{\:from\:both\:sides}$

$x^2+6x-\left(x^2+6x\right)=x^2+6x+9-\left(x^2+6x\right)$

$\mathrm{Simplify}$

$0=9$

$\mathrm{The\:sides\:are\:not\:equal}$

Thus, there is NO SOLUTION.

• We know that an equation that has no solution, such as $x = x + 2$, is called a contradiction.

Therefore, the equation $x\left(x\:+\:6\right)\:=\:\left(x\:+\:3\right)^2$ has NO SOLUTION, and is called contradiction.