Solve 75^(5)=2*x+2*x+2*x*2*x

$$75 ^ { ( 5 ) } = 2 \cdot x + 2 \cdot x + 2 \cdot x \cdot 2 \cdot x$$

Solve for x

$$x=\sqrt{593261719}-\frac{1}{2}\approx 24356.464486569$$

$$x=-\sqrt{593261719}-\frac{1}{2}\approx -24357.464486569$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$75^{5}=2x+2x+2x^{2}\times 2$$

Calculate 75 to the power of 5 and get 2373046875.

$$2373046875=2x+2x+2x^{2}\times 2$$

Combine 2x and 2x to get 4x.

$$2373046875=4x+2x^{2}\times 2$$

Multiply 2 and 2 to get 4.

$$2373046875=4x+4x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4x+4x^{2}=2373046875$$

Subtract 2373046875 from both sides.

$$4x+4x^{2}-2373046875=0$$

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.

$$4x^{2}+4x-2373046875=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 4 for b, and -2373046875 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 4\left(-2373046875\right)}}{2\times 4}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 4\left(-2373046875\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-4±\sqrt{16-16\left(-2373046875\right)}}{2\times 4}$$

Multiply -16 times -2373046875.

$$x=\frac{-4±\sqrt{16+37968750000}}{2\times 4}$$

Add 16 to 37968750000.

$$x=\frac{-4±\sqrt{37968750016}}{2\times 4}$$

Take the square root of 37968750016.

$$x=\frac{-4±8\sqrt{593261719}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-4±8\sqrt{593261719}}{8}$$

Now solve the equation x=\frac{-4±8\sqrt{593261719}}{8} when ± is plus. Add -4 to 8\sqrt{593261719}.

$$x=\frac{8\sqrt{593261719}-4}{8}$$

Divide -4+8\sqrt{593261719} by 8.

$$x=\sqrt{593261719}-\frac{1}{2}$$

Now solve the equation x=\frac{-4±8\sqrt{593261719}}{8} when ± is minus. Subtract 8\sqrt{593261719} from -4.

$$x=\frac{-8\sqrt{593261719}-4}{8}$$

Divide -4-8\sqrt{593261719} by 8.

$$x=-\sqrt{593261719}-\frac{1}{2}$$

The equation is now solved.

$$x=\sqrt{593261719}-\frac{1}{2} x=-\sqrt{593261719}-\frac{1}{2}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$75^{5}=2x+2x+2x^{2}\times 2$$

Calculate 75 to the power of 5 and get 2373046875.

$$2373046875=2x+2x+2x^{2}\times 2$$

Combine 2x and 2x to get 4x.

$$2373046875=4x+2x^{2}\times 2$$

Multiply 2 and 2 to get 4.

$$2373046875=4x+4x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4x+4x^{2}=2373046875$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4x^{2}+4x=2373046875$$

Divide both sides by 4.

$$\frac{4x^{2}+4x}{4}=\frac{2373046875}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{4}{4}x=\frac{2373046875}{4}$$

Divide 4 by 4.

$$x^{2}+x=\frac{2373046875}{4}$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{2373046875}{4}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{2373046875+1}{4}$$

Add \frac{2373046875}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=593261719$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=593261719$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{593261719}$$

Simplify.

$$x+\frac{1}{2}=\sqrt{593261719} x+\frac{1}{2}=-\sqrt{593261719}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\sqrt{593261719}-\frac{1}{2} x=-\sqrt{593261719}-\frac{1}{2}$$