Solve {left(sqrt{2018+x}+sqrt{2017+x}right)}^2=4

$${ \left( \sqrt{ 2018+x } + \sqrt{ 2017+x } \right) }^{ 2 } =4$$

Solve for x

$$x = -\frac{32263}{16} = -2016\frac{7}{16} = -2016.4375$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2018+x}+\sqrt{2017+x}\right)^{2}.

$$\left(\sqrt{2018+x}\right)^{2}+2\sqrt{2018+x}\sqrt{2017+x}+\left(\sqrt{2017+x}\right)^{2}=4$$

Calculate \sqrt{2018+x} to the power of 2 and get 2018+x.

$$2018+x+2\sqrt{2018+x}\sqrt{2017+x}+\left(\sqrt{2017+x}\right)^{2}=4$$

Calculate \sqrt{2017+x} to the power of 2 and get 2017+x.

$$2018+x+2\sqrt{2018+x}\sqrt{2017+x}+2017+x=4$$

Add 2018 and 2017 to get 4035.

$$4035+x+2\sqrt{2018+x}\sqrt{2017+x}+x=4$$

Combine x and x to get 2x.

$$4035+2x+2\sqrt{2018+x}\sqrt{2017+x}=4$$

Subtract 4035 from both sides.

$$2x+2\sqrt{2018+x}\sqrt{2017+x}=4-4035$$

Subtract 4035 from 4 to get -4031.

$$2x+2\sqrt{2018+x}\sqrt{2017+x}=-4031$$

Subtract 2x from both sides of the equation.

$$2\sqrt{2018+x}\sqrt{2017+x}=-4031-2x$$

Square both sides of the equation.

$$\left(2\sqrt{2018+x}\sqrt{2017+x}\right)^{2}=\left(-2x-4031\right)^{2}$$

Expand \left(2\sqrt{2018+x}\sqrt{2017+x}\right)^{2}.

$$2^{2}\left(\sqrt{2018+x}\right)^{2}\left(\sqrt{2017+x}\right)^{2}=\left(-2x-4031\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{2018+x}\right)^{2}\left(\sqrt{2017+x}\right)^{2}=\left(-2x-4031\right)^{2}$$

Calculate \sqrt{2018+x} to the power of 2 and get 2018+x.

$$4\left(2018+x\right)\left(\sqrt{2017+x}\right)^{2}=\left(-2x-4031\right)^{2}$$

Calculate \sqrt{2017+x} to the power of 2 and get 2017+x.

$$4\left(2018+x\right)\left(2017+x\right)=\left(-2x-4031\right)^{2}$$

Use the distributive property to multiply 4 by 2018+x.

$$\left(8072+4x\right)\left(2017+x\right)=\left(-2x-4031\right)^{2}$$

Use the distributive property to multiply 8072+4x by 2017+x and combine like terms.

$$16281224+16140x+4x^{2}=\left(-2x-4031\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2x-4031\right)^{2}.

$$16281224+16140x+4x^{2}=4x^{2}+16124x+16248961$$

Subtract 4x^{2} from both sides.

$$16281224+16140x+4x^{2}-4x^{2}=16124x+16248961$$

Combine 4x^{2} and -4x^{2} to get 0.

$$16281224+16140x=16124x+16248961$$

Subtract 16124x from both sides.

$$16281224+16140x-16124x=16248961$$

Combine 16140x and -16124x to get 16x.

$$16281224+16x=16248961$$

Subtract 16281224 from both sides.

$$16x=16248961-16281224$$

Subtract 16281224 from 16248961 to get -32263.

$$16x=-32263$$

Divide both sides by 16.

$$x=\frac{-32263}{16}$$

Fraction \frac{-32263}{16} can be rewritten as -\frac{32263}{16} by extracting the negative sign.

$$x=-\frac{32263}{16}$$

Substitute -\frac{32263}{16} for x in the equation \left(\sqrt{2018+x}+\sqrt{2017+x}\right)^{2}=4.

$$\left(\sqrt{2018-\frac{32263}{16}}+\sqrt{2017-\frac{32263}{16}}\right)^{2}=4$$

Simplify. The value x=-\frac{32263}{16} satisfies the equation.

$$4=4$$

Equation 2\sqrt{x+2017}\sqrt{x+2018}=-2x-4031 has a unique solution.

$$x=-\frac{32263}{16}$$