Solve for x
x = \frac{\sqrt{13} + 1}{2} \approx 2.302775638
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x^{2}=\left(\sqrt{x+3}\right)^{2}
Square both sides of the equation.
x^{2}=x+3
Calculate \sqrt{x+3} to the power of 2 and get x+3.
x^{2}-x=3
Subtract x from both sides.
x^{2}-x-3=0
Subtract 3 from both sides.
x=\frac{-\left(-1\right)±\sqrt{1-4\left(-3\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1+12}}{2}
Multiply -4 times -3.
x=\frac{-\left(-1\right)±\sqrt{13}}{2}
Add 1 to 12.
x=\frac{1±\sqrt{13}}{2}
The opposite of -1 is 1.
x=\frac{\sqrt{13}+1}{2}
Now solve the equation x=\frac{1±\sqrt{13}}{2} when ± is plus. Add 1 to \sqrt{13}.
x=\frac{1-\sqrt{13}}{2}
Now solve the equation x=\frac{1±\sqrt{13}}{2} when ± is minus. Subtract \sqrt{13} from 1.
x=\frac{\sqrt{13}+1}{2} x=\frac{1-\sqrt{13}}{2}
The equation is now solved.
\frac{\sqrt{13}+1}{2}=\sqrt{\frac{\sqrt{13}+1}{2}+3}
Substitute \frac{\sqrt{13}+1}{2} for x in the equation x=\sqrt{x+3}.
\frac{1}{2}\times 13^{\frac{1}{2}}+\frac{1}{2}=\frac{1}{2}+\frac{1}{2}\times 13^{\frac{1}{2}}
Simplify. The value x=\frac{\sqrt{13}+1}{2} satisfies the equation.
\frac{1-\sqrt{13}}{2}=\sqrt{\frac{1-\sqrt{13}}{2}+3}
Substitute \frac{1-\sqrt{13}}{2} for x in the equation x=\sqrt{x+3}.
\frac{1}{2}-\frac{1}{2}\times 13^{\frac{1}{2}}=-\left(\frac{1}{2}-\frac{1}{2}\times 13^{\frac{1}{2}}\right)
Simplify. The value x=\frac{1-\sqrt{13}}{2} does not satisfy the equation because the left and the right hand side have opposite signs.
x=\frac{\sqrt{13}+1}{2}
Equation x=\sqrt{x+3} has a unique solution.
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