Solve x-7sqrt{x+4}=14

$$x - 7 \sqrt { x + 4 } = 14$$

Solve for x

$$x=77$$

Solution Steps

Subtract x from both sides of the equation.

$$-7\sqrt{x+4}=14-x$$

Square both sides of the equation.

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

Expand \left(-7\sqrt{x+4}\right)^{2}.

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

Calculate -7 to the power of 2 and get 49.

$$49\left(\sqrt{x+4}\right)^{2}=\left(14-x\right)^{2}$$

Calculate \sqrt{x+4} to the power of 2 and get x+4.

$$49\left(x+4\right)=\left(14-x\right)^{2}$$

Use the distributive property to multiply 49 by x+4.

$$49x+196=\left(14-x\right)^{2}$$

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

$$49x+196=196-28x+x^{2}$$

Subtract 196 from both sides.

$$49x+196-196=-28x+x^{2}$$

Subtract 196 from 196 to get 0.

$$49x=-28x+x^{2}$$

Add 28x to both sides.

$$49x+28x=x^{2}$$

Combine 49x and 28x to get 77x.

$$77x=x^{2}$$

Subtract x^{2} from both sides.

$$77x-x^{2}=0$$

Factor out x.

$$x\left(77-x\right)=0$$

To find equation solutions, solve x=0 and 77-x=0.

$$x=0 x=77$$

Substitute 0 for x in the equation x-7\sqrt{x+4}=14.

$$0-7\sqrt{0+4}=14$$

Simplify. The value x=0 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-14=14$$

Substitute 77 for x in the equation x-7\sqrt{x+4}=14.

$$77-7\sqrt{77+4}=14$$

Simplify. The value x=77 satisfies the equation.

$$14=14$$

Equation -7\sqrt{x+4}=14-x has a unique solution.

$$x=77$$