Solve 4x-6-4x+4-x+1=2sqrt{4x+5}

$$4 x - 6 - 4 x + 4 - x + 1 = 2 \sqrt { 4 x + 5 }$$

Solve for x

$$x=7-2\sqrt{17}\approx -1.246211251$$

Solution Steps

Square both sides of the equation.

$$\left(4x-6-4x+4-x+1\right)^{2}=\left(2\sqrt{4x+5}\right)^{2}$$

Combine 4x and -4x to get 0.

$$\left(-6+4-x+1\right)^{2}=\left(2\sqrt{4x+5}\right)^{2}$$

Add -6 and 4 to get -2.

$$\left(-2-x+1\right)^{2}=\left(2\sqrt{4x+5}\right)^{2}$$

Add -2 and 1 to get -1.

$$\left(-1-x\right)^{2}=\left(2\sqrt{4x+5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-1-x\right)^{2}.

$$1+2x+x^{2}=\left(2\sqrt{4x+5}\right)^{2}$$

Expand \left(2\sqrt{4x+5}\right)^{2}.

$$1+2x+x^{2}=2^{2}\left(\sqrt{4x+5}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$1+2x+x^{2}=4\left(\sqrt{4x+5}\right)^{2}$$

Calculate \sqrt{4x+5} to the power of 2 and get 4x+5.

$$1+2x+x^{2}=4\left(4x+5\right)$$

Use the distributive property to multiply 4 by 4x+5.

$$1+2x+x^{2}=16x+20$$

Subtract 16x from both sides.

$$1+2x+x^{2}-16x=20$$

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

$$1-14x+x^{2}=20$$

Subtract 20 from both sides.

$$1-14x+x^{2}-20=0$$

Subtract 20 from 1 to get -19.

$$-19-14x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-14x-19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-19\right)}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-19\right)}}{2}$$

Multiply -4 times -19.

$$x=\frac{-\left(-14\right)±\sqrt{196+76}}{2}$$

Add 196 to 76.

$$x=\frac{-\left(-14\right)±\sqrt{272}}{2}$$

Take the square root of 272.

$$x=\frac{-\left(-14\right)±4\sqrt{17}}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±4\sqrt{17}}{2}$$

Now solve the equation x=\frac{14±4\sqrt{17}}{2} when ± is plus. Add 14 to 4\sqrt{17}.

$$x=\frac{4\sqrt{17}+14}{2}$$

Divide 14+4\sqrt{17} by 2.

$$x=2\sqrt{17}+7$$

Now solve the equation x=\frac{14±4\sqrt{17}}{2} when ± is minus. Subtract 4\sqrt{17} from 14.

$$x=\frac{14-4\sqrt{17}}{2}$$

Divide 14-4\sqrt{17} by 2.

$$x=7-2\sqrt{17}$$

The equation is now solved.

$$x=2\sqrt{17}+7 x=7-2\sqrt{17}$$

Substitute 2\sqrt{17}+7 for x in the equation 4x-6-4x+4-x+1=2\sqrt{4x+5}.

$$4\left(2\sqrt{17}+7\right)-6-4\left(2\sqrt{17}+7\right)+4-\left(2\sqrt{17}+7\right)+1=2\sqrt{4\left(2\sqrt{17}+7\right)+5}$$

Simplify. The value x=2\sqrt{17}+7 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2\times 17^{\frac{1}{2}}-8=2\times 17^{\frac{1}{2}}+8$$

Substitute 7-2\sqrt{17} for x in the equation 4x-6-4x+4-x+1=2\sqrt{4x+5}.

$$4\left(7-2\sqrt{17}\right)-6-4\left(7-2\sqrt{17}\right)+4-\left(7-2\sqrt{17}\right)+1=2\sqrt{4\left(7-2\sqrt{17}\right)+5}$$

Simplify. The value x=7-2\sqrt{17} satisfies the equation.

$$-8+2\times 17^{\frac{1}{2}}=2\times 17^{\frac{1}{2}}-8$$

Equation -x-1=2\sqrt{4x+5} has a unique solution.

$$x=7-2\sqrt{17}$$