Question 201519
Solve for x:
{{{x = sqrt(x-1)+3}}} First, subtract 3 from both sides of the equation.
{{{x-3 = sqrt(x-1)}}} Next, square both sides. Note that you may introduce an extraneous root when you square both sides so you must check your solutions to confirm their validity.
{{{(x-3)^2 = x-1}}} Simplify the left side.
{{{x^2-6x+9 = x-1}}} Subtract x from both sides.
{{{x^2-7x+9 = -1}}} Now add 1 to both sides.
{{{x^2-7x+10 = 0}}} Solve this quadratic equation by factoring.
{{{(x-2)(x-5) = 0}}} Apply the zero product rule.
{{{x-2 = 0}}} or {{{x-5 = 0}}} so that...
{{{highlight_green(x = 2)}}} or {{{highlight_green(x = 5)}}}
Check the solutions:
{{{x = sqrt(x-1)+3}}} Substitute x = 2.
{{{2 = sqrt(2-1)+3}}}
{{{2 = sqrt(1)+3}}} Note that {{{sqrt(1) = 1}}} or {{{sqrt(1) = -1}}}
{{{2 = 1+3}}} or {{{2 = -1+3}}}
{{{highlight(2 <> 4)}}} or {{{highlight_green(2 = 2)}}} ...and for the other solution...
{{{x = sqrt(x-1)+3}}} Substitute x = 5
{{{5 = sqrt(5-1)+3}}}
{{{5 = sqrt(4)+3}}}
{{{5 = 2+3}}} or {{{5 = -2+3}}}
{{{highlight_green(5 = 5)}}} or {{{highlight(5 <> 1)}}}