Solve left(25xright)^2+20x+4=0

$${ \left(25x \right) }^{ 2 } +20x+4 = 0$$

Solve for x (complex solution)

$$x=\frac{-2+4\sqrt{6}i}{125}\approx -0.016+0.078383672i$$

$$x=\frac{-4\sqrt{6}i-2}{125}\approx -0.016-0.078383672i$$

Steps Using the Quadratic Formula

Expand \left(25x\right)^{2}.

$$25^{2}x^{2}+20x+4=0$$

Calculate 25 to the power of 2 and get 625.

$$625x^{2}+20x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 625 for a, 20 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 625\times 4}}{2\times 625}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 625\times 4}}{2\times 625}$$

Multiply -4 times 625.

$$x=\frac{-20±\sqrt{400-2500\times 4}}{2\times 625}$$

Multiply -2500 times 4.

$$x=\frac{-20±\sqrt{400-10000}}{2\times 625}$$

Add 400 to -10000.

$$x=\frac{-20±\sqrt{-9600}}{2\times 625}$$

Take the square root of -9600.

$$x=\frac{-20±40\sqrt{6}i}{2\times 625}$$

Multiply 2 times 625.

$$x=\frac{-20±40\sqrt{6}i}{1250}$$

Now solve the equation x=\frac{-20±40\sqrt{6}i}{1250} when ± is plus. Add -20 to 40i\sqrt{6}.

$$x=\frac{-20+40\sqrt{6}i}{1250}$$

Divide -20+40i\sqrt{6} by 1250.

$$x=\frac{-2+4\sqrt{6}i}{125}$$

Now solve the equation x=\frac{-20±40\sqrt{6}i}{1250} when ± is minus. Subtract 40i\sqrt{6} from -20.

$$x=\frac{-40\sqrt{6}i-20}{1250}$$

Divide -20-40i\sqrt{6} by 1250.

$$x=\frac{-4\sqrt{6}i-2}{125}$$

The equation is now solved.

$$x=\frac{-2+4\sqrt{6}i}{125} x=\frac{-4\sqrt{6}i-2}{125}$$

Steps for Completing the Square

Expand \left(25x\right)^{2}.

$$25^{2}x^{2}+20x+4=0$$

Calculate 25 to the power of 2 and get 625.

$$625x^{2}+20x+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$625x^{2}+20x=-4$$

Divide both sides by 625.

$$\frac{625x^{2}+20x}{625}=-\frac{4}{625}$$

Dividing by 625 undoes the multiplication by 625.

$$x^{2}+\frac{20}{625}x=-\frac{4}{625}$$

Reduce the fraction \frac{20}{625} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{4}{125}x=-\frac{4}{625}$$

Divide \frac{4}{125}, the coefficient of the x term, by 2 to get \frac{2}{125}. Then add the square of \frac{2}{125} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{125}x+\left(\frac{2}{125}\right)^{2}=-\frac{4}{625}+\left(\frac{2}{125}\right)^{2}$$

Square \frac{2}{125} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{125}x+\frac{4}{15625}=-\frac{4}{625}+\frac{4}{15625}$$

Add -\frac{4}{625} to \frac{4}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{125}x+\frac{4}{15625}=-\frac{96}{15625}$$

Factor x^{2}+\frac{4}{125}x+\frac{4}{15625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{125}\right)^{2}=-\frac{96}{15625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{125}\right)^{2}}=\sqrt{-\frac{96}{15625}}$$

Simplify.

$$x+\frac{2}{125}=\frac{4\sqrt{6}i}{125} x+\frac{2}{125}=-\frac{4\sqrt{6}i}{125}$$

Subtract \frac{2}{125} from both sides of the equation.

$$x=\frac{-2+4\sqrt{6}i}{125} x=\frac{-4\sqrt{6}i-2}{125}$$