Solve 625x^2+196x-1054=0

$$625 { x }^{ 2 } +196x-1054=0$$

Solve for x

$$x = \frac{\sqrt{668354} - 98}{625} \approx 1.151246727$$

$$x=\frac{-\sqrt{668354}-98}{625}\approx -1.464846727$$

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.

$$625x^{2}+196x-1054=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 625 for a, 196 for b, and -1054 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-196±\sqrt{196^{2}-4\times 625\left(-1054\right)}}{2\times 625}$$

Square 196.

$$x=\frac{-196±\sqrt{38416-4\times 625\left(-1054\right)}}{2\times 625}$$

Multiply -4 times 625.

$$x=\frac{-196±\sqrt{38416-2500\left(-1054\right)}}{2\times 625}$$

Multiply -2500 times -1054.

$$x=\frac{-196±\sqrt{38416+2635000}}{2\times 625}$$

Add 38416 to 2635000.

$$x=\frac{-196±\sqrt{2673416}}{2\times 625}$$

Take the square root of 2673416.

$$x=\frac{-196±2\sqrt{668354}}{2\times 625}$$

Multiply 2 times 625.

$$x=\frac{-196±2\sqrt{668354}}{1250}$$

Now solve the equation x=\frac{-196±2\sqrt{668354}}{1250} when ± is plus. Add -196 to 2\sqrt{668354}.

$$x=\frac{2\sqrt{668354}-196}{1250}$$

Divide -196+2\sqrt{668354} by 1250.

$$x=\frac{\sqrt{668354}-98}{625}$$

Now solve the equation x=\frac{-196±2\sqrt{668354}}{1250} when ± is minus. Subtract 2\sqrt{668354} from -196.

$$x=\frac{-2\sqrt{668354}-196}{1250}$$

Divide -196-2\sqrt{668354} by 1250.

$$x=\frac{-\sqrt{668354}-98}{625}$$

The equation is now solved.

$$x=\frac{\sqrt{668354}-98}{625} x=\frac{-\sqrt{668354}-98}{625}$$

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.

$$625x^{2}+196x-1054=0$$

Add 1054 to both sides of the equation.

$$625x^{2}+196x-1054-\left(-1054\right)=-\left(-1054\right)$$

Subtracting -1054 from itself leaves 0.

$$625x^{2}+196x=-\left(-1054\right)$$

Subtract -1054 from 0.

$$625x^{2}+196x=1054$$

Divide both sides by 625.

$$\frac{625x^{2}+196x}{625}=\frac{1054}{625}$$

Dividing by 625 undoes the multiplication by 625.

$$x^{2}+\frac{196}{625}x=\frac{1054}{625}$$

Divide \frac{196}{625}, the coefficient of the x term, by 2 to get \frac{98}{625}. Then add the square of \frac{98}{625} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{196}{625}x+\left(\frac{98}{625}\right)^{2}=\frac{1054}{625}+\left(\frac{98}{625}\right)^{2}$$

Square \frac{98}{625} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{196}{625}x+\frac{9604}{390625}=\frac{1054}{625}+\frac{9604}{390625}$$

Add \frac{1054}{625} to \frac{9604}{390625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{196}{625}x+\frac{9604}{390625}=\frac{668354}{390625}$$

Factor x^{2}+\frac{196}{625}x+\frac{9604}{390625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{98}{625}\right)^{2}=\frac{668354}{390625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{98}{625}\right)^{2}}=\sqrt{\frac{668354}{390625}}$$

Simplify.

$$x+\frac{98}{625}=\frac{\sqrt{668354}}{625} x+\frac{98}{625}=-\frac{\sqrt{668354}}{625}$$

Subtract \frac{98}{625} from both sides of the equation.

$$x=\frac{\sqrt{668354}-98}{625} x=\frac{-\sqrt{668354}-98}{625}$$