Solve 2sqrt{x}+2sqrt{x+7}=14

$$2 \sqrt { x } + 2 \sqrt { x + 7 } = 14$$

Solve for x

$$x=9$$

Solution Steps

Subtract 2\sqrt{x+7} from both sides of the equation.

$$2\sqrt{x}=14-2\sqrt{x+7}$$

Square both sides of the equation.

$$\left(2\sqrt{x}\right)^{2}=\left(14-2\sqrt{x+7}\right)^{2}$$

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

$$2^{2}\left(\sqrt{x}\right)^{2}=\left(14-2\sqrt{x+7}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x}\right)^{2}=\left(14-2\sqrt{x+7}\right)^{2}$$

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

$$4x=\left(14-2\sqrt{x+7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(14-2\sqrt{x+7}\right)^{2}.

$$4x=196-56\sqrt{x+7}+4\left(\sqrt{x+7}\right)^{2}$$

Calculate \sqrt{x+7} to the power of 2 and get x+7.

$$4x=196-56\sqrt{x+7}+4\left(x+7\right)$$

Use the distributive property to multiply 4 by x+7.

$$4x=196-56\sqrt{x+7}+4x+28$$

Add 196 and 28 to get 224.

$$4x=224-56\sqrt{x+7}+4x$$

Add 56\sqrt{x+7} to both sides.

$$4x+56\sqrt{x+7}=224+4x$$

Subtract 4x from both sides.

$$4x+56\sqrt{x+7}-4x=224$$

Combine 4x and -4x to get 0.

$$56\sqrt{x+7}=224$$

Divide both sides by 56.

$$\sqrt{x+7}=\frac{224}{56}$$

Divide 224 by 56 to get 4.

$$\sqrt{x+7}=4$$

Square both sides of the equation.

$$x+7=16$$

Subtract 7 from both sides of the equation.

$$x+7-7=16-7$$

Subtracting 7 from itself leaves 0.

$$x=16-7$$

Subtract 7 from 16.

$$x=9$$

Substitute 9 for x in the equation 2\sqrt{x}+2\sqrt{x+7}=14.

$$2\sqrt{9}+2\sqrt{9+7}=14$$

Simplify. The value x=9 satisfies the equation.

$$14=14$$

Equation 2\sqrt{x}=-2\sqrt{x+7}+14 has a unique solution.

$$x=9$$