Solve sqrt{x^2+2x+3}=2sqrt{2x^2+4x}

$$\sqrt { x ^ { 2 } + 2 x + 3 } = 2 \sqrt { 2 x ^ { 2 } + 4 x }$$

Solve for x

$$x=\frac{\sqrt{70}}{7}-1\approx 0.195228609$$

$$x=-\frac{\sqrt{70}}{7}-1\approx -2.195228609$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x^{2}+2x+3}\right)^{2}=\left(2\sqrt{2x^{2}+4x}\right)^{2}$$

Calculate \sqrt{x^{2}+2x+3} to the power of 2 and get x^{2}+2x+3.

$$x^{2}+2x+3=\left(2\sqrt{2x^{2}+4x}\right)^{2}$$

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

$$x^{2}+2x+3=2^{2}\left(\sqrt{2x^{2}+4x}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}+2x+3=4\left(\sqrt{2x^{2}+4x}\right)^{2}$$

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

$$x^{2}+2x+3=4\left(2x^{2}+4x\right)$$

Use the distributive property to multiply 4 by 2x^{2}+4x.

$$x^{2}+2x+3=8x^{2}+16x$$

Subtract 8x^{2} from both sides.

$$x^{2}+2x+3-8x^{2}=16x$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}+2x+3=16x$$

Subtract 16x from both sides.

$$-7x^{2}+2x+3-16x=0$$

Combine 2x and -16x to get -14x.

$$-7x^{2}-14x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, -14 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-7\right)\times 3}}{2\left(-7\right)}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-7\right)\times 3}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-14\right)±\sqrt{196+28\times 3}}{2\left(-7\right)}$$

Multiply 28 times 3.

$$x=\frac{-\left(-14\right)±\sqrt{196+84}}{2\left(-7\right)}$$

Add 196 to 84.

$$x=\frac{-\left(-14\right)±\sqrt{280}}{2\left(-7\right)}$$

Take the square root of 280.

$$x=\frac{-\left(-14\right)±2\sqrt{70}}{2\left(-7\right)}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{70}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{14±2\sqrt{70}}{-14}$$

Now solve the equation x=\frac{14±2\sqrt{70}}{-14} when ± is plus. Add 14 to 2\sqrt{70}.

$$x=\frac{2\sqrt{70}+14}{-14}$$

Divide 14+2\sqrt{70} by -14.

$$x=-\frac{\sqrt{70}}{7}-1$$

Now solve the equation x=\frac{14±2\sqrt{70}}{-14} when ± is minus. Subtract 2\sqrt{70} from 14.

$$x=\frac{14-2\sqrt{70}}{-14}$$

Divide 14-2\sqrt{70} by -14.

$$x=\frac{\sqrt{70}}{7}-1$$

The equation is now solved.

$$x=-\frac{\sqrt{70}}{7}-1 x=\frac{\sqrt{70}}{7}-1$$

Substitute -\frac{\sqrt{70}}{7}-1 for x in the equation \sqrt{x^{2}+2x+3}=2\sqrt{2x^{2}+4x}.

$$\sqrt{\left(-\frac{\sqrt{70}}{7}-1\right)^{2}+2\left(-\frac{\sqrt{70}}{7}-1\right)+3}=2\sqrt{2\left(-\frac{\sqrt{70}}{7}-1\right)^{2}+4\left(-\frac{\sqrt{70}}{7}-1\right)}$$

Simplify. The value x=-\frac{\sqrt{70}}{7}-1 satisfies the equation.

$$\frac{2}{7}\times 42^{\frac{1}{2}}=\frac{2}{7}\times 42^{\frac{1}{2}}$$

Substitute \frac{\sqrt{70}}{7}-1 for x in the equation \sqrt{x^{2}+2x+3}=2\sqrt{2x^{2}+4x}.

$$\sqrt{\left(\frac{\sqrt{70}}{7}-1\right)^{2}+2\left(\frac{\sqrt{70}}{7}-1\right)+3}=2\sqrt{2\left(\frac{\sqrt{70}}{7}-1\right)^{2}+4\left(\frac{\sqrt{70}}{7}-1\right)}$$

Simplify. The value x=\frac{\sqrt{70}}{7}-1 satisfies the equation.

$$\frac{2}{7}\times 42^{\frac{1}{2}}=\frac{2}{7}\times 42^{\frac{1}{2}}$$

List all solutions of \sqrt{x^{2}+2x+3}=2\sqrt{2x^{2}+4x}.

$$x=-\frac{\sqrt{70}}{7}-1 x=\frac{\sqrt{70}}{7}-1$$