Solve x=sqrt{5x+14}

$$x = \sqrt { 5 x + 14 }$$

Solve for x

$$x=7$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(\sqrt{5x+14}\right)^{2}$$

Calculate \sqrt{5x+14} to the power of 2 and get 5x+14.

$$x^{2}=5x+14$$

Subtract 5x from both sides.

$$x^{2}-5x=14$$

Subtract 14 from both sides.

$$x^{2}-5x-14=0$$

To solve the equation, factor x^{2}-5x-14 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-5 ab=-14$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -14.

$$1,-14 2,-7$$

Calculate the sum for each pair.

$$1-14=-13 2-7=-5$$

The solution is the pair that gives sum -5.

$$a=-7 b=2$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-7\right)\left(x+2\right)$$

To find equation solutions, solve x-7=0 and x+2=0.

$$x=7 x=-2$$

Substitute 7 for x in the equation x=\sqrt{5x+14}.

$$7=\sqrt{5\times 7+14}$$

Simplify. The value x=7 satisfies the equation.

$$7=7$$

Substitute -2 for x in the equation x=\sqrt{5x+14}.

$$-2=\sqrt{5\left(-2\right)+14}$$

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

Equation x=\sqrt{5x+14} has a unique solution.

$$x=7$$