Solve 25x^2-102x+2^2=0

$$25 x ^ { 2 } - 102 x + 2 ^ { 2 } = 0$$

Solve for x

$$x = \frac{\sqrt{2501} + 51}{25} \approx 4.04039996$$

$$x=\frac{51-\sqrt{2501}}{25}\approx 0.03960004$$

Steps Using the Quadratic Formula

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-102x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -102 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-102\right)±\sqrt{\left(-102\right)^{2}-4\times 25\times 4}}{2\times 25}$$

Square -102.

$$x=\frac{-\left(-102\right)±\sqrt{10404-4\times 25\times 4}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-102\right)±\sqrt{10404-100\times 4}}{2\times 25}$$

Multiply -100 times 4.

$$x=\frac{-\left(-102\right)±\sqrt{10404-400}}{2\times 25}$$

Add 10404 to -400.

$$x=\frac{-\left(-102\right)±\sqrt{10004}}{2\times 25}$$

Take the square root of 10004.

$$x=\frac{-\left(-102\right)±2\sqrt{2501}}{2\times 25}$$

The opposite of -102 is 102.

$$x=\frac{102±2\sqrt{2501}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{102±2\sqrt{2501}}{50}$$

Now solve the equation x=\frac{102±2\sqrt{2501}}{50} when ± is plus. Add 102 to 2\sqrt{2501}.

$$x=\frac{2\sqrt{2501}+102}{50}$$

Divide 102+2\sqrt{2501} by 50.

$$x=\frac{\sqrt{2501}+51}{25}$$

Now solve the equation x=\frac{102±2\sqrt{2501}}{50} when ± is minus. Subtract 2\sqrt{2501} from 102.

$$x=\frac{102-2\sqrt{2501}}{50}$$

Divide 102-2\sqrt{2501} by 50.

$$x=\frac{51-\sqrt{2501}}{25}$$

The equation is now solved.

$$x=\frac{\sqrt{2501}+51}{25} x=\frac{51-\sqrt{2501}}{25}$$

Steps for Completing the Square

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-102x+4=0$$

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

$$25x^{2}-102x=-4$$

Divide both sides by 25.

$$\frac{25x^{2}-102x}{25}=-\frac{4}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}-\frac{102}{25}x=-\frac{4}{25}$$

Divide -\frac{102}{25}, the coefficient of the x term, by 2 to get -\frac{51}{25}. Then add the square of -\frac{51}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{102}{25}x+\left(-\frac{51}{25}\right)^{2}=-\frac{4}{25}+\left(-\frac{51}{25}\right)^{2}$$

Square -\frac{51}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{102}{25}x+\frac{2601}{625}=-\frac{4}{25}+\frac{2601}{625}$$

Add -\frac{4}{25} to \frac{2601}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{102}{25}x+\frac{2601}{625}=\frac{2501}{625}$$

Factor x^{2}-\frac{102}{25}x+\frac{2601}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{51}{25}\right)^{2}=\frac{2501}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{51}{25}\right)^{2}}=\sqrt{\frac{2501}{625}}$$

Simplify.

$$x-\frac{51}{25}=\frac{\sqrt{2501}}{25} x-\frac{51}{25}=-\frac{\sqrt{2501}}{25}$$

Add \frac{51}{25} to both sides of the equation.

$$x=\frac{\sqrt{2501}+51}{25} x=\frac{51-\sqrt{2501}}{25}$$