Solve x=4.25x^2+635x-39075

$$x=4.25 { x }^{ 2 } +635x-39075$$

Solve for x

$$x = \frac{2 \sqrt{1066231} - 1268}{17} \approx 46.89230838$$

$$x=\frac{-2\sqrt{1066231}-1268}{17}\approx -196.068778968$$

Steps Using the Quadratic Formula

Subtract 4.25x^{2} from both sides.

$$x-4.25x^{2}=635x-39075$$

Subtract 635x from both sides.

$$x-4.25x^{2}-635x=-39075$$

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

$$-634x-4.25x^{2}=-39075$$

Add 39075 to both sides.

$$-634x-4.25x^{2}+39075=0$$

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.

$$-4.25x^{2}-634x+39075=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.25 for a, -634 for b, and 39075 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-634\right)±\sqrt{\left(-634\right)^{2}-4\left(-4.25\right)\times 39075}}{2\left(-4.25\right)}$$

Square -634.

$$x=\frac{-\left(-634\right)±\sqrt{401956-4\left(-4.25\right)\times 39075}}{2\left(-4.25\right)}$$

Multiply -4 times -4.25.

$$x=\frac{-\left(-634\right)±\sqrt{401956+17\times 39075}}{2\left(-4.25\right)}$$

Multiply 17 times 39075.

$$x=\frac{-\left(-634\right)±\sqrt{401956+664275}}{2\left(-4.25\right)}$$

Add 401956 to 664275.

$$x=\frac{-\left(-634\right)±\sqrt{1066231}}{2\left(-4.25\right)}$$

The opposite of -634 is 634.

$$x=\frac{634±\sqrt{1066231}}{2\left(-4.25\right)}$$

Multiply 2 times -4.25.

$$x=\frac{634±\sqrt{1066231}}{-8.5}$$

Now solve the equation x=\frac{634±\sqrt{1066231}}{-8.5} when ± is plus. Add 634 to \sqrt{1066231}.

$$x=\frac{\sqrt{1066231}+634}{-8.5}$$

Divide 634+\sqrt{1066231} by -8.5 by multiplying 634+\sqrt{1066231} by the reciprocal of -8.5.

$$x=\frac{-2\sqrt{1066231}-1268}{17}$$

Now solve the equation x=\frac{634±\sqrt{1066231}}{-8.5} when ± is minus. Subtract \sqrt{1066231} from 634.

$$x=\frac{634-\sqrt{1066231}}{-8.5}$$

Divide 634-\sqrt{1066231} by -8.5 by multiplying 634-\sqrt{1066231} by the reciprocal of -8.5.

$$x=\frac{2\sqrt{1066231}-1268}{17}$$

The equation is now solved.

$$x=\frac{-2\sqrt{1066231}-1268}{17} x=\frac{2\sqrt{1066231}-1268}{17}$$

Steps for Completing the Square

Subtract 4.25x^{2} from both sides.

$$x-4.25x^{2}=635x-39075$$

Subtract 635x from both sides.

$$x-4.25x^{2}-635x=-39075$$

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

$$-634x-4.25x^{2}=-39075$$

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.

$$-4.25x^{2}-634x=-39075$$

Divide both sides of the equation by -4.25, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-4.25x^{2}-634x}{-4.25}=-\frac{39075}{-4.25}$$

Dividing by -4.25 undoes the multiplication by -4.25.

$$x^{2}+\left(-\frac{634}{-4.25}\right)x=-\frac{39075}{-4.25}$$

Divide -634 by -4.25 by multiplying -634 by the reciprocal of -4.25.

$$x^{2}+\frac{2536}{17}x=-\frac{39075}{-4.25}$$

Divide -39075 by -4.25 by multiplying -39075 by the reciprocal of -4.25.

$$x^{2}+\frac{2536}{17}x=\frac{156300}{17}$$

Divide \frac{2536}{17}, the coefficient of the x term, by 2 to get \frac{1268}{17}. Then add the square of \frac{1268}{17} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2536}{17}x+\frac{1268}{17}^{2}=\frac{156300}{17}+\frac{1268}{17}^{2}$$

Square \frac{1268}{17} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2536}{17}x+\frac{1607824}{289}=\frac{156300}{17}+\frac{1607824}{289}$$

Add \frac{156300}{17} to \frac{1607824}{289} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2536}{17}x+\frac{1607824}{289}=\frac{4264924}{289}$$

Factor x^{2}+\frac{2536}{17}x+\frac{1607824}{289}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1268}{17}\right)^{2}=\frac{4264924}{289}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1268}{17}\right)^{2}}=\sqrt{\frac{4264924}{289}}$$

Simplify.

$$x+\frac{1268}{17}=\frac{2\sqrt{1066231}}{17} x+\frac{1268}{17}=-\frac{2\sqrt{1066231}}{17}$$

Subtract \frac{1268}{17} from both sides of the equation.

$$x=\frac{2\sqrt{1066231}-1268}{17} x=\frac{-2\sqrt{1066231}-1268}{17}$$