Solve x+27/2=sqrt{27*x}+6

$$\frac{ x+27 }{ 2 } = \sqrt{ 27 \times x } +6$$

Solve for x

$$x=75$$

$$x=3$$

Solution Steps

Subtract 6 from both sides of the equation.

$$\frac{x+27}{2}-6=\sqrt{27x}$$

Multiply both sides of the equation by 2.

$$x+27-12=2\sqrt{27x}$$

Subtract 12 from 27 to get 15.

$$x+15=2\sqrt{27x}$$

Square both sides of the equation.

$$\left(x+15\right)^{2}=\left(2\sqrt{27x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+15\right)^{2}.

$$x^{2}+30x+225=\left(2\sqrt{27x}\right)^{2}$$

Expand \left(2\sqrt{27x}\right)^{2}.

$$x^{2}+30x+225=2^{2}\left(\sqrt{27x}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}+30x+225=4\left(\sqrt{27x}\right)^{2}$$

Calculate \sqrt{27x} to the power of 2 and get 27x.

$$x^{2}+30x+225=4\times 27x$$

Multiply 4 and 27 to get 108.

$$x^{2}+30x+225=108x$$

Subtract 108x from both sides.

$$x^{2}+30x+225-108x=0$$

Combine 30x and -108x to get -78x.

$$x^{2}-78x+225=0$$

To solve the equation, factor x^{2}-78x+225 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.

$$a+b=-78 ab=225$$

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 225.

$$-1,-225 -3,-75 -5,-45 -9,-25 -15,-15$$

Calculate the sum for each pair.

$$-1-225=-226 -3-75=-78 -5-45=-50 -9-25=-34 -15-15=-30$$

The solution is the pair that gives sum -78.

$$a=-75 b=-3$$

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

$$\left(x-75\right)\left(x-3\right)$$

To find equation solutions, solve x-75=0 and x-3=0.

$$x=75 x=3$$

Substitute 75 for x in the equation \frac{x+27}{2}=\sqrt{27x}+6.

$$\frac{75+27}{2}=\sqrt{27\times 75}+6$$

Simplify. The value x=75 satisfies the equation.

$$51=51$$

Substitute 3 for x in the equation \frac{x+27}{2}=\sqrt{27x}+6.

$$\frac{3+27}{2}=\sqrt{27\times 3}+6$$

Simplify. The value x=3 satisfies the equation.

$$15=15$$

List all solutions of x+15=2\sqrt{27x}.

$$x=75 x=3$$