Solve x=4+sqrt{4x-19}

$$x = 4 + \sqrt { 4 x - 19 }$$

Solve for x

$$x=7$$

$$x=5$$

Solution Steps

Subtract 4 from both sides of the equation.

$$x-4=\sqrt{4x-19}$$

Square both sides of the equation.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16=\left(\sqrt{4x-19}\right)^{2}$$

Calculate \sqrt{4x-19} to the power of 2 and get 4x-19.

$$x^{2}-8x+16=4x-19$$

Subtract 4x from both sides.

$$x^{2}-8x+16-4x=-19$$

Combine -8x and -4x to get -12x.

$$x^{2}-12x+16=-19$$

Add 19 to both sides.

$$x^{2}-12x+16+19=0$$

Add 16 and 19 to get 35.

$$x^{2}-12x+35=0$$

To solve the equation, factor x^{2}-12x+35 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.

$$a+b=-12 ab=35$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 35.

$$-1,-35 -5,-7$$

Calculate the sum for each pair.

$$-1-35=-36 -5-7=-12$$

The solution is the pair that gives sum -12.

$$a=-7 b=-5$$

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

$$\left(x-7\right)\left(x-5\right)$$

To find equation solutions, solve x-7=0 and x-5=0.

$$x=7 x=5$$

Substitute 7 for x in the equation x=4+\sqrt{4x-19}.

$$7=4+\sqrt{4\times 7-19}$$

Simplify. The value x=7 satisfies the equation.

$$7=7$$

Substitute 5 for x in the equation x=4+\sqrt{4x-19}.

$$5=4+\sqrt{4\times 5-19}$$

Simplify. The value x=5 satisfies the equation.

$$5=5$$

List all solutions of x-4=\sqrt{4x-19}.

$$x=7 x=5$$