Answer
$\color{blue}{x=-1, x=3}$
Work Step by Step
Square both sides to obtain:
$4x+13=(2x-1)^2
\\4x+13=(2x)^2-2(2x)(1)+(-1)^2
\\4x+13=4x^2-4x+1$
Move all terms to the left side of the equation.
Note that when a term is transferred to the other side of the equation, its sign changes to its opposite.
$0=4x^2-4x+1-4x-13
\\0=4x^2-8x-12
\\4x^2-8x-12=0$
Factor out $4$ to obtain:
$4(x^2-2x-3)=0$
Factor the trinomial to obtain:
$4(x-3)(x+1)=0$
Divide $4$ to both sides to obtain:
$(x-3)(x+1)=0$
Use the zero-factor property by equating each factor to zero.
$x-3=0$ or $x+1=0$
Solve each equation to obtain:
$x=3$ or $x=-1$.
Upon checking, both proposed solutions satisfy the original equation.
Thus, the solutions to the given equation are $\color{blue}{x=-1, x=3}$.
