Solve sqrt{2x+3}=4-sqrt{x}

$$\sqrt { 2 x + 3 } = 4 - \sqrt { x }$$

Solve for x

$$x=45-8\sqrt{29}\approx 1.918681543$$

Solution Steps

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{x}\right)^{2}.

$$2x+3=16-8\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$2x+3=16-8\sqrt{x}+x$$

Subtract 16+x from both sides of the equation.

$$2x+3-\left(16+x\right)=-8\sqrt{x}$$

To find the opposite of 16+x, find the opposite of each term.

$$2x+3-16-x=-8\sqrt{x}$$

Subtract 16 from 3 to get -13.

$$2x-13-x=-8\sqrt{x}$$

Combine 2x and -x to get x.

$$x-13=-8\sqrt{x}$$

Square both sides of the equation.

$$\left(x-13\right)^{2}=\left(-8\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-13\right)^{2}.

$$x^{2}-26x+169=\left(-8\sqrt{x}\right)^{2}$$

Expand \left(-8\sqrt{x}\right)^{2}.

$$x^{2}-26x+169=\left(-8\right)^{2}\left(\sqrt{x}\right)^{2}$$

Calculate -8 to the power of 2 and get 64.

$$x^{2}-26x+169=64\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}-26x+169=64x$$

Subtract 64x from both sides.

$$x^{2}-26x+169-64x=0$$

Combine -26x and -64x to get -90x.

$$x^{2}-90x+169=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -90 for b, and 169 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 169}}{2}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 169}}{2}$$

Multiply -4 times 169.

$$x=\frac{-\left(-90\right)±\sqrt{8100-676}}{2}$$

Add 8100 to -676.

$$x=\frac{-\left(-90\right)±\sqrt{7424}}{2}$$

Take the square root of 7424.

$$x=\frac{-\left(-90\right)±16\sqrt{29}}{2}$$

The opposite of -90 is 90.

$$x=\frac{90±16\sqrt{29}}{2}$$

Now solve the equation x=\frac{90±16\sqrt{29}}{2} when ± is plus. Add 90 to 16\sqrt{29}.

$$x=\frac{16\sqrt{29}+90}{2}$$

Divide 90+16\sqrt{29} by 2.

$$x=8\sqrt{29}+45$$

Now solve the equation x=\frac{90±16\sqrt{29}}{2} when ± is minus. Subtract 16\sqrt{29} from 90.

$$x=\frac{90-16\sqrt{29}}{2}$$

Divide 90-16\sqrt{29} by 2.

$$x=45-8\sqrt{29}$$

The equation is now solved.

$$x=8\sqrt{29}+45 x=45-8\sqrt{29}$$

Substitute 8\sqrt{29}+45 for x in the equation \sqrt{2x+3}=4-\sqrt{x}.

$$\sqrt{2\left(8\sqrt{29}+45\right)+3}=4-\sqrt{8\sqrt{29}+45}$$

Simplify. The value x=8\sqrt{29}+45 does not satisfy the equation because the left and the right hand side have opposite signs.

$$8+29^{\frac{1}{2}}=-29^{\frac{1}{2}}$$

Substitute 45-8\sqrt{29} for x in the equation \sqrt{2x+3}=4-\sqrt{x}.

$$\sqrt{2\left(45-8\sqrt{29}\right)+3}=4-\sqrt{45-8\sqrt{29}}$$

Simplify. The value x=45-8\sqrt{29} satisfies the equation.

$$8-29^{\frac{1}{2}}=8-29^{\frac{1}{2}}$$

Equation \sqrt{2x+3}=-\sqrt{x}+4 has a unique solution.

$$x=45-8\sqrt{29}$$