Solve x=425x^2+635x-39075

$$x=425 { x }^{ 2 } +635x-39075$$

Solve for x

$$x = \frac{2 \sqrt{4176841} - 317}{425} \approx 8.87168059$$

$$x=\frac{-2\sqrt{4176841}-317}{425}\approx -10.363445296$$

Steps Using the Quadratic Formula

Subtract 425x^{2} from both sides.

$$x-425x^{2}=635x-39075$$

Subtract 635x from both sides.

$$x-425x^{2}-635x=-39075$$

Combine x and -635x to get -634x.

$$-634x-425x^{2}=-39075$$

Add 39075 to both sides.

$$-634x-425x^{2}+39075=0$$

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.

$$-425x^{2}-634x+39075=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -425 for a, -634 for b, and 39075 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-634\right)±\sqrt{\left(-634\right)^{2}-4\left(-425\right)\times 39075}}{2\left(-425\right)}$$

Square -634.

$$x=\frac{-\left(-634\right)±\sqrt{401956-4\left(-425\right)\times 39075}}{2\left(-425\right)}$$

Multiply -4 times -425.

$$x=\frac{-\left(-634\right)±\sqrt{401956+1700\times 39075}}{2\left(-425\right)}$$

Multiply 1700 times 39075.

$$x=\frac{-\left(-634\right)±\sqrt{401956+66427500}}{2\left(-425\right)}$$

Add 401956 to 66427500.

$$x=\frac{-\left(-634\right)±\sqrt{66829456}}{2\left(-425\right)}$$

Take the square root of 66829456.

$$x=\frac{-\left(-634\right)±4\sqrt{4176841}}{2\left(-425\right)}$$

The opposite of -634 is 634.

$$x=\frac{634±4\sqrt{4176841}}{2\left(-425\right)}$$

Multiply 2 times -425.

$$x=\frac{634±4\sqrt{4176841}}{-850}$$

Now solve the equation x=\frac{634±4\sqrt{4176841}}{-850} when ± is plus. Add 634 to 4\sqrt{4176841}.

$$x=\frac{4\sqrt{4176841}+634}{-850}$$

Divide 634+4\sqrt{4176841} by -850.

$$x=\frac{-2\sqrt{4176841}-317}{425}$$

Now solve the equation x=\frac{634±4\sqrt{4176841}}{-850} when ± is minus. Subtract 4\sqrt{4176841} from 634.

$$x=\frac{634-4\sqrt{4176841}}{-850}$$

Divide 634-4\sqrt{4176841} by -850.

$$x=\frac{2\sqrt{4176841}-317}{425}$$

The equation is now solved.

$$x=\frac{-2\sqrt{4176841}-317}{425} x=\frac{2\sqrt{4176841}-317}{425}$$

Steps for Completing the Square

Subtract 425x^{2} from both sides.

$$x-425x^{2}=635x-39075$$

Subtract 635x from both sides.

$$x-425x^{2}-635x=-39075$$

Combine x and -635x to get -634x.

$$-634x-425x^{2}=-39075$$

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.

$$-425x^{2}-634x=-39075$$

Divide both sides by -425.

$$\frac{-425x^{2}-634x}{-425}=-\frac{39075}{-425}$$

Dividing by -425 undoes the multiplication by -425.

$$x^{2}+\left(-\frac{634}{-425}\right)x=-\frac{39075}{-425}$$

Divide -634 by -425.

$$x^{2}+\frac{634}{425}x=-\frac{39075}{-425}$$

Reduce the fraction \frac{-39075}{-425} to lowest terms by extracting and canceling out 25.

$$x^{2}+\frac{634}{425}x=\frac{1563}{17}$$

Divide \frac{634}{425}, the coefficient of the x term, by 2 to get \frac{317}{425}. Then add the square of \frac{317}{425} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{634}{425}x+\left(\frac{317}{425}\right)^{2}=\frac{1563}{17}+\left(\frac{317}{425}\right)^{2}$$

Square \frac{317}{425} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{634}{425}x+\frac{100489}{180625}=\frac{1563}{17}+\frac{100489}{180625}$$

Add \frac{1563}{17} to \frac{100489}{180625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{634}{425}x+\frac{100489}{180625}=\frac{16707364}{180625}$$

Factor x^{2}+\frac{634}{425}x+\frac{100489}{180625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{317}{425}\right)^{2}=\frac{16707364}{180625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{317}{425}\right)^{2}}=\sqrt{\frac{16707364}{180625}}$$

Simplify.

$$x+\frac{317}{425}=\frac{2\sqrt{4176841}}{425} x+\frac{317}{425}=-\frac{2\sqrt{4176841}}{425}$$

Subtract \frac{317}{425} from both sides of the equation.

$$x=\frac{2\sqrt{4176841}-317}{425} x=\frac{-2\sqrt{4176841}-317}{425}$$