Solve 676x^2-5625x-10000=0

$$676 { x }^{ 2 } -5625x-10000=0$$

Solve for x

$$x = \frac{25 \sqrt{93889} + 5625}{1352} \approx 9.826426032$$

$$x=\frac{5625-25\sqrt{93889}}{1352}\approx -1.505420115$$

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.

$$676x^{2}-5625x-10000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 676 for a, -5625 for b, and -10000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5625\right)±\sqrt{\left(-5625\right)^{2}-4\times 676\left(-10000\right)}}{2\times 676}$$

Square -5625.

$$x=\frac{-\left(-5625\right)±\sqrt{31640625-4\times 676\left(-10000\right)}}{2\times 676}$$

Multiply -4 times 676.

$$x=\frac{-\left(-5625\right)±\sqrt{31640625-2704\left(-10000\right)}}{2\times 676}$$

Multiply -2704 times -10000.

$$x=\frac{-\left(-5625\right)±\sqrt{31640625+27040000}}{2\times 676}$$

Add 31640625 to 27040000.

$$x=\frac{-\left(-5625\right)±\sqrt{58680625}}{2\times 676}$$

Take the square root of 58680625.

$$x=\frac{-\left(-5625\right)±25\sqrt{93889}}{2\times 676}$$

The opposite of -5625 is 5625.

$$x=\frac{5625±25\sqrt{93889}}{2\times 676}$$

Multiply 2 times 676.

$$x=\frac{5625±25\sqrt{93889}}{1352}$$

Now solve the equation x=\frac{5625±25\sqrt{93889}}{1352} when ± is plus. Add 5625 to 25\sqrt{93889}.

$$x=\frac{25\sqrt{93889}+5625}{1352}$$

Now solve the equation x=\frac{5625±25\sqrt{93889}}{1352} when ± is minus. Subtract 25\sqrt{93889} from 5625.

$$x=\frac{5625-25\sqrt{93889}}{1352}$$

The equation is now solved.

$$x=\frac{25\sqrt{93889}+5625}{1352} x=\frac{5625-25\sqrt{93889}}{1352}$$

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.

$$676x^{2}-5625x-10000=0$$

Add 10000 to both sides of the equation.

$$676x^{2}-5625x-10000-\left(-10000\right)=-\left(-10000\right)$$

Subtracting -10000 from itself leaves 0.

$$676x^{2}-5625x=-\left(-10000\right)$$

Subtract -10000 from 0.

$$676x^{2}-5625x=10000$$

Divide both sides by 676.

$$\frac{676x^{2}-5625x}{676}=\frac{10000}{676}$$

Dividing by 676 undoes the multiplication by 676.

$$x^{2}-\frac{5625}{676}x=\frac{10000}{676}$$

Reduce the fraction \frac{10000}{676} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{5625}{676}x=\frac{2500}{169}$$

Divide -\frac{5625}{676}, the coefficient of the x term, by 2 to get -\frac{5625}{1352}. Then add the square of -\frac{5625}{1352} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5625}{676}x+\left(-\frac{5625}{1352}\right)^{2}=\frac{2500}{169}+\left(-\frac{5625}{1352}\right)^{2}$$

Square -\frac{5625}{1352} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5625}{676}x+\frac{31640625}{1827904}=\frac{2500}{169}+\frac{31640625}{1827904}$$

Add \frac{2500}{169} to \frac{31640625}{1827904} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5625}{676}x+\frac{31640625}{1827904}=\frac{58680625}{1827904}$$

Factor x^{2}-\frac{5625}{676}x+\frac{31640625}{1827904}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5625}{1352}\right)^{2}=\frac{58680625}{1827904}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5625}{1352}\right)^{2}}=\sqrt{\frac{58680625}{1827904}}$$

Simplify.

$$x-\frac{5625}{1352}=\frac{25\sqrt{93889}}{1352} x-\frac{5625}{1352}=-\frac{25\sqrt{93889}}{1352}$$

Add \frac{5625}{1352} to both sides of the equation.

$$x=\frac{25\sqrt{93889}+5625}{1352} x=\frac{5625-25\sqrt{93889}}{1352}$$