Solve 1/4x^2-25x+25=0

$$\frac{ 1 }{ 4 } { x }^{ 2 } -25x+25=0$$

Solve for x

$$x=20\sqrt{6}+50\approx 98.989794856$$

$$x=50-20\sqrt{6}\approx 1.010205144$$

Steps Using the Quadratic Formula

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.

$$\frac{1}{4}x^{2}-25x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{4} for a, -25 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times \frac{1}{4}\times 25}}{2\times \frac{1}{4}}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times \frac{1}{4}\times 25}}{2\times \frac{1}{4}}$$

Multiply -4 times \frac{1}{4}.

$$x=\frac{-\left(-25\right)±\sqrt{625-25}}{2\times \frac{1}{4}}$$

Add 625 to -25.

$$x=\frac{-\left(-25\right)±\sqrt{600}}{2\times \frac{1}{4}}$$

Take the square root of 600.

$$x=\frac{-\left(-25\right)±10\sqrt{6}}{2\times \frac{1}{4}}$$

The opposite of -25 is 25.

$$x=\frac{25±10\sqrt{6}}{2\times \frac{1}{4}}$$

Multiply 2 times \frac{1}{4}.

$$x=\frac{25±10\sqrt{6}}{\frac{1}{2}}$$

Now solve the equation x=\frac{25±10\sqrt{6}}{\frac{1}{2}} when ± is plus. Add 25 to 10\sqrt{6}.

$$x=\frac{10\sqrt{6}+25}{\frac{1}{2}}$$

Divide 25+10\sqrt{6} by \frac{1}{2} by multiplying 25+10\sqrt{6} by the reciprocal of \frac{1}{2}.

$$x=20\sqrt{6}+50$$

Now solve the equation x=\frac{25±10\sqrt{6}}{\frac{1}{2}} when ± is minus. Subtract 10\sqrt{6} from 25.

$$x=\frac{25-10\sqrt{6}}{\frac{1}{2}}$$

Divide 25-10\sqrt{6} by \frac{1}{2} by multiplying 25-10\sqrt{6} by the reciprocal of \frac{1}{2}.

$$x=50-20\sqrt{6}$$

The equation is now solved.

$$x=20\sqrt{6}+50 x=50-20\sqrt{6}$$

Steps for Completing the Square

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.

$$\frac{1}{4}x^{2}-25x+25=0$$

Subtract 25 from both sides of the equation.

$$\frac{1}{4}x^{2}-25x+25-25=-25$$

Subtracting 25 from itself leaves 0.

$$\frac{1}{4}x^{2}-25x=-25$$

Multiply both sides by 4.

$$\frac{\frac{1}{4}x^{2}-25x}{\frac{1}{4}}=-\frac{25}{\frac{1}{4}}$$

Dividing by \frac{1}{4} undoes the multiplication by \frac{1}{4}.

$$x^{2}+\left(-\frac{25}{\frac{1}{4}}\right)x=-\frac{25}{\frac{1}{4}}$$

Divide -25 by \frac{1}{4} by multiplying -25 by the reciprocal of \frac{1}{4}.

$$x^{2}-100x=-\frac{25}{\frac{1}{4}}$$

Divide -25 by \frac{1}{4} by multiplying -25 by the reciprocal of \frac{1}{4}.

$$x^{2}-100x=-100$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-100+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-100+2500$$

Add -100 to 2500.

$$x^{2}-100x+2500=2400$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=2400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2400}$$

Simplify.

$$x-50=20\sqrt{6} x-50=-20\sqrt{6}$$

Add 50 to both sides of the equation.

$$x=20\sqrt{6}+50 x=50-20\sqrt{6}$$