Solve 625x^2+1570x-10362=0

$$625 { x }^{ 2 } +1570x-10362=0$$

Solve for x

$$x = \frac{\sqrt{283699} - 157}{125} \approx 3.005072166$$

$$x=\frac{-\sqrt{283699}-157}{125}\approx -5.517072166$$

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.

$$625x^{2}+1570x-10362=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 625 for a, 1570 for b, and -10362 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1570±\sqrt{1570^{2}-4\times 625\left(-10362\right)}}{2\times 625}$$

Square 1570.

$$x=\frac{-1570±\sqrt{2464900-4\times 625\left(-10362\right)}}{2\times 625}$$

Multiply -4 times 625.

$$x=\frac{-1570±\sqrt{2464900-2500\left(-10362\right)}}{2\times 625}$$

Multiply -2500 times -10362.

$$x=\frac{-1570±\sqrt{2464900+25905000}}{2\times 625}$$

Add 2464900 to 25905000.

$$x=\frac{-1570±\sqrt{28369900}}{2\times 625}$$

Take the square root of 28369900.

$$x=\frac{-1570±10\sqrt{283699}}{2\times 625}$$

Multiply 2 times 625.

$$x=\frac{-1570±10\sqrt{283699}}{1250}$$

Now solve the equation x=\frac{-1570±10\sqrt{283699}}{1250} when ± is plus. Add -1570 to 10\sqrt{283699}.

$$x=\frac{10\sqrt{283699}-1570}{1250}$$

Divide -1570+10\sqrt{283699} by 1250.

$$x=\frac{\sqrt{283699}-157}{125}$$

Now solve the equation x=\frac{-1570±10\sqrt{283699}}{1250} when ± is minus. Subtract 10\sqrt{283699} from -1570.

$$x=\frac{-10\sqrt{283699}-1570}{1250}$$

Divide -1570-10\sqrt{283699} by 1250.

$$x=\frac{-\sqrt{283699}-157}{125}$$

The equation is now solved.

$$x=\frac{\sqrt{283699}-157}{125} x=\frac{-\sqrt{283699}-157}{125}$$

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.

$$625x^{2}+1570x-10362=0$$

Add 10362 to both sides of the equation.

$$625x^{2}+1570x-10362-\left(-10362\right)=-\left(-10362\right)$$

Subtracting -10362 from itself leaves 0.

$$625x^{2}+1570x=-\left(-10362\right)$$

Subtract -10362 from 0.

$$625x^{2}+1570x=10362$$

Divide both sides by 625.

$$\frac{625x^{2}+1570x}{625}=\frac{10362}{625}$$

Dividing by 625 undoes the multiplication by 625.

$$x^{2}+\frac{1570}{625}x=\frac{10362}{625}$$

Reduce the fraction \frac{1570}{625} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{314}{125}x=\frac{10362}{625}$$

Divide \frac{314}{125}, the coefficient of the x term, by 2 to get \frac{157}{125}. Then add the square of \frac{157}{125} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{314}{125}x+\left(\frac{157}{125}\right)^{2}=\frac{10362}{625}+\left(\frac{157}{125}\right)^{2}$$

Square \frac{157}{125} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{314}{125}x+\frac{24649}{15625}=\frac{10362}{625}+\frac{24649}{15625}$$

Add \frac{10362}{625} to \frac{24649}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{314}{125}x+\frac{24649}{15625}=\frac{283699}{15625}$$

Factor x^{2}+\frac{314}{125}x+\frac{24649}{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}.

$$\left(x+\frac{157}{125}\right)^{2}=\frac{283699}{15625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{157}{125}\right)^{2}}=\sqrt{\frac{283699}{15625}}$$

Simplify.

$$x+\frac{157}{125}=\frac{\sqrt{283699}}{125} x+\frac{157}{125}=-\frac{\sqrt{283699}}{125}$$

Subtract \frac{157}{125} from both sides of the equation.

$$x=\frac{\sqrt{283699}-157}{125} x=\frac{-\sqrt{283699}-157}{125}$$