Solve sqrt{x-1}-2=2sqrt{x+3}

$$\sqrt { x - 1 } - 2 = 2 \sqrt { x + 3 }$$

Solve for x (complex solution)

$$x=\frac{-16\sqrt{2}i-19}{9}\approx -2.111111111-2.514157444i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x-1}-2\right)^{2}=\left(2\sqrt{x+3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{x-1}-2\right)^{2}.

$$\left(\sqrt{x-1}\right)^{2}-4\sqrt{x-1}+4=\left(2\sqrt{x+3}\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x-1-4\sqrt{x-1}+4=\left(2\sqrt{x+3}\right)^{2}$$

Add -1 and 4 to get 3.

$$x+3-4\sqrt{x-1}=\left(2\sqrt{x+3}\right)^{2}$$

Expand \left(2\sqrt{x+3}\right)^{2}.

$$x+3-4\sqrt{x-1}=2^{2}\left(\sqrt{x+3}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x+3-4\sqrt{x-1}=4\left(\sqrt{x+3}\right)^{2}$$

Calculate \sqrt{x+3} to the power of 2 and get x+3.

$$x+3-4\sqrt{x-1}=4\left(x+3\right)$$

Use the distributive property to multiply 4 by x+3.

$$x+3-4\sqrt{x-1}=4x+12$$

Subtract x+3 from both sides of the equation.

$$-4\sqrt{x-1}=4x+12-\left(x+3\right)$$

To find the opposite of x+3, find the opposite of each term.

$$-4\sqrt{x-1}=4x+12-x-3$$

Combine 4x and -x to get 3x.

$$-4\sqrt{x-1}=3x+12-3$$

Subtract 3 from 12 to get 9.

$$-4\sqrt{x-1}=3x+9$$

Square both sides of the equation.

$$\left(-4\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

Expand \left(-4\sqrt{x-1}\right)^{2}.

$$\left(-4\right)^{2}\left(\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

Calculate -4 to the power of 2 and get 16.

$$16\left(\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$16\left(x-1\right)=\left(3x+9\right)^{2}$$

Use the distributive property to multiply 16 by x-1.

$$16x-16=\left(3x+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+9\right)^{2}.

$$16x-16=9x^{2}+54x+81$$

Subtract 9x^{2} from both sides.

$$16x-16-9x^{2}=54x+81$$

Subtract 54x from both sides.

$$16x-16-9x^{2}-54x=81$$

Combine 16x and -54x to get -38x.

$$-38x-16-9x^{2}=81$$

Subtract 81 from both sides.

$$-38x-16-9x^{2}-81=0$$

Subtract 81 from -16 to get -97.

$$-38x-97-9x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-9x^{2}-38x-97=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -38 for b, and -97 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\left(-9\right)\left(-97\right)}}{2\left(-9\right)}$$

Square -38.

$$x=\frac{-\left(-38\right)±\sqrt{1444-4\left(-9\right)\left(-97\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-38\right)±\sqrt{1444+36\left(-97\right)}}{2\left(-9\right)}$$

Multiply 36 times -97.

$$x=\frac{-\left(-38\right)±\sqrt{1444-3492}}{2\left(-9\right)}$$

Add 1444 to -3492.

$$x=\frac{-\left(-38\right)±\sqrt{-2048}}{2\left(-9\right)}$$

Take the square root of -2048.

$$x=\frac{-\left(-38\right)±32\sqrt{2}i}{2\left(-9\right)}$$

The opposite of -38 is 38.

$$x=\frac{38±32\sqrt{2}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{38±32\sqrt{2}i}{-18}$$

Now solve the equation x=\frac{38±32\sqrt{2}i}{-18} when ± is plus. Add 38 to 32i\sqrt{2}.

$$x=\frac{38+32\sqrt{2}i}{-18}$$

Divide 38+32i\sqrt{2} by -18.

$$x=\frac{-16\sqrt{2}i-19}{9}$$

Now solve the equation x=\frac{38±32\sqrt{2}i}{-18} when ± is minus. Subtract 32i\sqrt{2} from 38.

$$x=\frac{-32\sqrt{2}i+38}{-18}$$

Divide 38-32i\sqrt{2} by -18.

$$x=\frac{-19+16\sqrt{2}i}{9}$$

The equation is now solved.

$$x=\frac{-16\sqrt{2}i-19}{9} x=\frac{-19+16\sqrt{2}i}{9}$$

Substitute \frac{-16\sqrt{2}i-19}{9} for x in the equation \sqrt{x-1}-2=2\sqrt{x+3}.

$$\sqrt{\frac{-16\sqrt{2}i-19}{9}-1}-2=2\sqrt{\frac{-16\sqrt{2}i-19}{9}+3}$$

Simplify. The value x=\frac{-16\sqrt{2}i-19}{9} satisfies the equation.

$$-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

Substitute \frac{-19+16\sqrt{2}i}{9} for x in the equation \sqrt{x-1}-2=2\sqrt{x+3}.

$$\sqrt{\frac{-19+16\sqrt{2}i}{9}-1}-2=2\sqrt{\frac{-19+16\sqrt{2}i}{9}+3}$$

Simplify. The value x=\frac{-19+16\sqrt{2}i}{9} does not satisfy the equation.

$$-\frac{4}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

Substitute \frac{-16\sqrt{2}i-19}{9} for x in the equation \sqrt{x-1}-2=2\sqrt{x+3}.

$$\sqrt{\frac{-16\sqrt{2}i-19}{9}-1}-2=2\sqrt{\frac{-16\sqrt{2}i-19}{9}+3}$$

Simplify. The value x=\frac{-16\sqrt{2}i-19}{9} satisfies the equation.

$$-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

Equation \sqrt{x-1}-2=2\sqrt{x+3} has a unique solution.

$$x=\frac{-16\sqrt{2}i-19}{9}$$