Solve sqrt{7x+2}+2x=0

$$\sqrt { 7 x + 2 } + 2 x = 0$$

Solve for x

$$x=-\frac{1}{4}=-0.25$$

Solution Steps

Subtract 2x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

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

Calculate -2 to the power of 2 and get 4.

$$7x+2=4x^{2}$$

Subtract 4x^{2} from both sides.

$$7x+2-4x^{2}=0$$

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

$$-4x^{2}+7x+2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=7 ab=-4\times 2=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 7.

$$a=8 b=-1$$

Rewrite -4x^{2}+7x+2 as \left(-4x^{2}+8x\right)+\left(-x+2\right).

$$\left(-4x^{2}+8x\right)+\left(-x+2\right)$$

Factor out 4x in -4x^{2}+8x.

$$4x\left(-x+2\right)-x+2$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(4x+1\right)$$

To find equation solutions, solve -x+2=0 and 4x+1=0.

$$x=2 x=-\frac{1}{4}$$

Substitute 2 for x in the equation \sqrt{7x+2}+2x=0.

$$\sqrt{7\times 2+2}+2\times 2=0$$

Simplify. The value x=2 does not satisfy the equation.

$$8=0$$

Substitute -\frac{1}{4} for x in the equation \sqrt{7x+2}+2x=0.

$$\sqrt{7\left(-\frac{1}{4}\right)+2}+2\left(-\frac{1}{4}\right)=0$$

Simplify. The value x=-\frac{1}{4} satisfies the equation.

$$0=0$$

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

$$x=-\frac{1}{4}$$