Solve sqrt{-49+14x}=x

$$\sqrt { - 49 + 14 x } = x$$

Solve for x

$$x=7$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{-49+14x} to the power of 2 and get -49+14x.

$$-49+14x=x^{2}$$

Subtract x^{2} from both sides.

$$-49+14x-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+14x-49=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-49. To find a and b, set up a system to be solved.

$$a+b=14 ab=-\left(-49\right)=49$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 49.

$$1,49 7,7$$

Calculate the sum for each pair.

$$1+49=50 7+7=14$$

The solution is the pair that gives sum 14.

$$a=7 b=7$$

Rewrite -x^{2}+14x-49 as \left(-x^{2}+7x\right)+\left(7x-49\right).

$$\left(-x^{2}+7x\right)+\left(7x-49\right)$$

Factor out -x in the first and 7 in the second group.

$$-x\left(x-7\right)+7\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x+7\right)$$

To find equation solutions, solve x-7=0 and -x+7=0.

$$x=7 x=7$$

Substitute 7 for x in the equation \sqrt{-49+14x}=x.

$$\sqrt{-49+14\times 7}=7$$

Simplify. The value x=7 satisfies the equation.

$$7=7$$

Substitute 7 for x in the equation \sqrt{-49+14x}=x.

$$\sqrt{-49+14\times 7}=7$$

Simplify. The value x=7 satisfies the equation.

$$7=7$$

List all solutions of \sqrt{14x-49}=x.

$$x=7 x=7$$