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

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

Solve for x

$$x=1$$

Solution Steps

Subtract -1 from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

$$5x+4=4x^{2}+4x+1$$

Subtract 4x^{2} from both sides.

$$5x+4-4x^{2}=4x+1$$

Subtract 4x from both sides.

$$5x+4-4x^{2}-4x=1$$

Combine 5x and -4x to get x.

$$x+4-4x^{2}=1$$

Subtract 1 from both sides.

$$x+4-4x^{2}-1=0$$

Subtract 1 from 4 to get 3.

$$x+3-4x^{2}=0$$

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

$$-4x^{2}+x+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=1 ab=-4\times 3=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 1.

$$a=4 b=-3$$

Rewrite -4x^{2}+x+3 as \left(-4x^{2}+4x\right)+\left(-3x+3\right).

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

Factor out 4x in the first and 3 in the second group.

$$4x\left(-x+1\right)+3\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(4x+3\right)$$

To find equation solutions, solve -x+1=0 and 4x+3=0.

$$x=1 x=-\frac{3}{4}$$

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

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

Simplify. The value x=1 satisfies the equation.

$$2=2$$

Substitute -\frac{3}{4} for x in the equation \sqrt{5x+4}-1=2x.

$$\sqrt{5\left(-\frac{3}{4}\right)+4}-1=2\left(-\frac{3}{4}\right)$$

Simplify. The value x=-\frac{3}{4} does not satisfy the equation.

$$-\frac{1}{2}=-\frac{3}{2}$$

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

$$x=1$$