Solve 43897+204x^2=-59414x^2+13216x+52929

$$43897+204 { x }^{ 2 } =-59414 { x }^{ 2 } +13216x+52929$$

Solve for x

$$x=\frac{2\sqrt{36383465}+3304}{29809}\approx 0.515540325$$

$$x=\frac{3304-2\sqrt{36383465}}{29809}\approx -0.293862308$$

Steps Using the Quadratic Formula

Add 59414x^{2} to both sides.

$$43897+204x^{2}+59414x^{2}=13216x+52929$$

Combine 204x^{2} and 59414x^{2} to get 59618x^{2}.

$$43897+59618x^{2}=13216x+52929$$

Subtract 13216x from both sides.

$$43897+59618x^{2}-13216x=52929$$

Subtract 52929 from both sides.

$$43897+59618x^{2}-13216x-52929=0$$

Subtract 52929 from 43897 to get -9032.

$$-9032+59618x^{2}-13216x=0$$

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.

$$59618x^{2}-13216x-9032=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 59618 for a, -13216 for b, and -9032 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13216\right)±\sqrt{\left(-13216\right)^{2}-4\times 59618\left(-9032\right)}}{2\times 59618}$$

Square -13216.

$$x=\frac{-\left(-13216\right)±\sqrt{174662656-4\times 59618\left(-9032\right)}}{2\times 59618}$$

Multiply -4 times 59618.

$$x=\frac{-\left(-13216\right)±\sqrt{174662656-238472\left(-9032\right)}}{2\times 59618}$$

Multiply -238472 times -9032.

$$x=\frac{-\left(-13216\right)±\sqrt{174662656+2153879104}}{2\times 59618}$$

Add 174662656 to 2153879104.

$$x=\frac{-\left(-13216\right)±\sqrt{2328541760}}{2\times 59618}$$

Take the square root of 2328541760.

$$x=\frac{-\left(-13216\right)±8\sqrt{36383465}}{2\times 59618}$$

The opposite of -13216 is 13216.

$$x=\frac{13216±8\sqrt{36383465}}{2\times 59618}$$

Multiply 2 times 59618.

$$x=\frac{13216±8\sqrt{36383465}}{119236}$$

Now solve the equation x=\frac{13216±8\sqrt{36383465}}{119236} when ± is plus. Add 13216 to 8\sqrt{36383465}.

$$x=\frac{8\sqrt{36383465}+13216}{119236}$$

Divide 13216+8\sqrt{36383465} by 119236.

$$x=\frac{2\sqrt{36383465}+3304}{29809}$$

Now solve the equation x=\frac{13216±8\sqrt{36383465}}{119236} when ± is minus. Subtract 8\sqrt{36383465} from 13216.

$$x=\frac{13216-8\sqrt{36383465}}{119236}$$

Divide 13216-8\sqrt{36383465} by 119236.

$$x=\frac{3304-2\sqrt{36383465}}{29809}$$

The equation is now solved.

$$x=\frac{2\sqrt{36383465}+3304}{29809} x=\frac{3304-2\sqrt{36383465}}{29809}$$

Steps for Completing the Square

Add 59414x^{2} to both sides.

$$43897+204x^{2}+59414x^{2}=13216x+52929$$

Combine 204x^{2} and 59414x^{2} to get 59618x^{2}.

$$43897+59618x^{2}=13216x+52929$$

Subtract 13216x from both sides.

$$43897+59618x^{2}-13216x=52929$$

Subtract 43897 from both sides.

$$59618x^{2}-13216x=52929-43897$$

Subtract 43897 from 52929 to get 9032.

$$59618x^{2}-13216x=9032$$

Divide both sides by 59618.

$$\frac{59618x^{2}-13216x}{59618}=\frac{9032}{59618}$$

Dividing by 59618 undoes the multiplication by 59618.

$$x^{2}+\left(-\frac{13216}{59618}\right)x=\frac{9032}{59618}$$

Reduce the fraction \frac{-13216}{59618} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{6608}{29809}x=\frac{9032}{59618}$$

Reduce the fraction \frac{9032}{59618} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{6608}{29809}x=\frac{4516}{29809}$$

Divide -\frac{6608}{29809}, the coefficient of the x term, by 2 to get -\frac{3304}{29809}. Then add the square of -\frac{3304}{29809} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6608}{29809}x+\left(-\frac{3304}{29809}\right)^{2}=\frac{4516}{29809}+\left(-\frac{3304}{29809}\right)^{2}$$

Square -\frac{3304}{29809} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6608}{29809}x+\frac{10916416}{888576481}=\frac{4516}{29809}+\frac{10916416}{888576481}$$

Add \frac{4516}{29809} to \frac{10916416}{888576481} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6608}{29809}x+\frac{10916416}{888576481}=\frac{145533860}{888576481}$$

Factor x^{2}-\frac{6608}{29809}x+\frac{10916416}{888576481}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3304}{29809}\right)^{2}=\frac{145533860}{888576481}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3304}{29809}\right)^{2}}=\sqrt{\frac{145533860}{888576481}}$$

Simplify.

$$x-\frac{3304}{29809}=\frac{2\sqrt{36383465}}{29809} x-\frac{3304}{29809}=-\frac{2\sqrt{36383465}}{29809}$$

Add \frac{3304}{29809} to both sides of the equation.

$$x=\frac{2\sqrt{36383465}+3304}{29809} x=\frac{3304-2\sqrt{36383465}}{29809}$$