Solve 25x=200000+2x^2+1*left(2xright)^2

$$25x=200000+2 { x }^{ 2 } +1 \times { \left(2x \right) }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{-25\sqrt{7679}i+25}{12}\approx 2.083333333-182.562299108i$$

$$x=\frac{25+25\sqrt{7679}i}{12}\approx 2.083333333+182.562299108i$$

Steps Using the Quadratic Formula

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

$$25x=200000+2x^{2}+1\times 2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$25x=200000+2x^{2}+1\times 4x^{2}$$

Multiply 1 and 4 to get 4.

$$25x=200000+2x^{2}+4x^{2}$$

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$25x=200000+6x^{2}$$

Subtract 200000 from both sides.

$$25x-200000=6x^{2}$$

Subtract 6x^{2} from both sides.

$$25x-200000-6x^{2}=0$$

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.

$$-6x^{2}+25x-200000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 25 for b, and -200000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-6\right)\left(-200000\right)}}{2\left(-6\right)}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\left(-6\right)\left(-200000\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-25±\sqrt{625+24\left(-200000\right)}}{2\left(-6\right)}$$

Multiply 24 times -200000.

$$x=\frac{-25±\sqrt{625-4800000}}{2\left(-6\right)}$$

Add 625 to -4800000.

$$x=\frac{-25±\sqrt{-4799375}}{2\left(-6\right)}$$

Take the square root of -4799375.

$$x=\frac{-25±25\sqrt{7679}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-25±25\sqrt{7679}i}{-12}$$

Now solve the equation x=\frac{-25±25\sqrt{7679}i}{-12} when ± is plus. Add -25 to 25i\sqrt{7679}.

$$x=\frac{-25+25\sqrt{7679}i}{-12}$$

Divide -25+25i\sqrt{7679} by -12.

$$x=\frac{-25\sqrt{7679}i+25}{12}$$

Now solve the equation x=\frac{-25±25\sqrt{7679}i}{-12} when ± is minus. Subtract 25i\sqrt{7679} from -25.

$$x=\frac{-25\sqrt{7679}i-25}{-12}$$

Divide -25-25i\sqrt{7679} by -12.

$$x=\frac{25+25\sqrt{7679}i}{12}$$

The equation is now solved.

$$x=\frac{-25\sqrt{7679}i+25}{12} x=\frac{25+25\sqrt{7679}i}{12}$$

Steps for Completing the Square

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

$$25x=200000+2x^{2}+1\times 2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$25x=200000+2x^{2}+1\times 4x^{2}$$

Multiply 1 and 4 to get 4.

$$25x=200000+2x^{2}+4x^{2}$$

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$25x=200000+6x^{2}$$

Subtract 6x^{2} from both sides.

$$25x-6x^{2}=200000$$

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.

$$-6x^{2}+25x=200000$$

Divide both sides by -6.

$$\frac{-6x^{2}+25x}{-6}=\frac{200000}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{25}{-6}x=\frac{200000}{-6}$$

Divide 25 by -6.

$$x^{2}-\frac{25}{6}x=\frac{200000}{-6}$$

Reduce the fraction \frac{200000}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{25}{6}x=-\frac{100000}{3}$$

Divide -\frac{25}{6}, the coefficient of the x term, by 2 to get -\frac{25}{12}. Then add the square of -\frac{25}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{6}x+\left(-\frac{25}{12}\right)^{2}=-\frac{100000}{3}+\left(-\frac{25}{12}\right)^{2}$$

Square -\frac{25}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{6}x+\frac{625}{144}=-\frac{100000}{3}+\frac{625}{144}$$

Add -\frac{100000}{3} to \frac{625}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{6}x+\frac{625}{144}=-\frac{4799375}{144}$$

Factor x^{2}-\frac{25}{6}x+\frac{625}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{12}\right)^{2}=-\frac{4799375}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{12}\right)^{2}}=\sqrt{-\frac{4799375}{144}}$$

Simplify.

$$x-\frac{25}{12}=\frac{25\sqrt{7679}i}{12} x-\frac{25}{12}=-\frac{25\sqrt{7679}i}{12}$$

Add \frac{25}{12} to both sides of the equation.

$$x=\frac{25+25\sqrt{7679}i}{12} x=\frac{-25\sqrt{7679}i+25}{12}$$