Solve 43.897+2.04x^2=-5.9414x^2+13.216x+52.929

$$43.897+2.04 { x }^{ 2 } =-5.9414 { x }^{ 2 } +13.216x+52.929$$

Solve for x

$$x=\frac{2\sqrt{723460430}}{39907}+\frac{4720}{5701}\approx 2.175920001$$

$$x=-\frac{2\sqrt{723460430}}{39907}+\frac{4720}{5701}\approx -0.52007015$$

Steps Using the Quadratic Formula

Add 5.9414x^{2} to both sides.

$$43.897+2.04x^{2}+5.9414x^{2}=13.216x+52.929$$

Combine 2.04x^{2} and 5.9414x^{2} to get 7.9814x^{2}.

$$43.897+7.9814x^{2}=13.216x+52.929$$

Subtract 13.216x from both sides.

$$43.897+7.9814x^{2}-13.216x=52.929$$

Subtract 52.929 from both sides.

$$43.897+7.9814x^{2}-13.216x-52.929=0$$

Subtract 52.929 from 43.897 to get -9.032.

$$-9.032+7.9814x^{2}-13.216x=0$$

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.

$$7.9814x^{2}-13.216x-9.032=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7.9814 for a, -13.216 for b, and -9.032 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13.216\right)±\sqrt{\left(-13.216\right)^{2}-4\times 7.9814\left(-9.032\right)}}{2\times 7.9814}$$

Square -13.216 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-13.216\right)±\sqrt{174.662656-4\times 7.9814\left(-9.032\right)}}{2\times 7.9814}$$

Multiply -4 times 7.9814.

$$x=\frac{-\left(-13.216\right)±\sqrt{174.662656-31.9256\left(-9.032\right)}}{2\times 7.9814}$$

Multiply -31.9256 times -9.032 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-13.216\right)±\sqrt{174.662656+288.3520192}}{2\times 7.9814}$$

Add 174.662656 to 288.3520192 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-13.216\right)±\sqrt{463.0146752}}{2\times 7.9814}$$

Take the square root of 463.0146752.

$$x=\frac{-\left(-13.216\right)±\frac{\sqrt{723460430}}{1250}}{2\times 7.9814}$$

The opposite of -13.216 is 13.216.

$$x=\frac{13.216±\frac{\sqrt{723460430}}{1250}}{2\times 7.9814}$$

Multiply 2 times 7.9814.

$$x=\frac{13.216±\frac{\sqrt{723460430}}{1250}}{15.9628}$$

Now solve the equation x=\frac{13.216±\frac{\sqrt{723460430}}{1250}}{15.9628} when ± is plus. Add 13.216 to \frac{\sqrt{723460430}}{1250}.

$$x=\frac{\frac{\sqrt{723460430}}{1250}+\frac{1652}{125}}{15.9628}$$

Divide \frac{1652}{125}+\frac{\sqrt{723460430}}{1250} by 15.9628 by multiplying \frac{1652}{125}+\frac{\sqrt{723460430}}{1250} by the reciprocal of 15.9628.

$$x=\frac{2\sqrt{723460430}}{39907}+\frac{4720}{5701}$$

Now solve the equation x=\frac{13.216±\frac{\sqrt{723460430}}{1250}}{15.9628} when ± is minus. Subtract \frac{\sqrt{723460430}}{1250} from 13.216.

$$x=\frac{-\frac{\sqrt{723460430}}{1250}+\frac{1652}{125}}{15.9628}$$

Divide \frac{1652}{125}-\frac{\sqrt{723460430}}{1250} by 15.9628 by multiplying \frac{1652}{125}-\frac{\sqrt{723460430}}{1250} by the reciprocal of 15.9628.

$$x=-\frac{2\sqrt{723460430}}{39907}+\frac{4720}{5701}$$

The equation is now solved.

$$x=\frac{2\sqrt{723460430}}{39907}+\frac{4720}{5701} x=-\frac{2\sqrt{723460430}}{39907}+\frac{4720}{5701}$$

Steps for Completing the Square

Add 5.9414x^{2} to both sides.

$$43.897+2.04x^{2}+5.9414x^{2}=13.216x+52.929$$

Combine 2.04x^{2} and 5.9414x^{2} to get 7.9814x^{2}.

$$43.897+7.9814x^{2}=13.216x+52.929$$

Subtract 13.216x from both sides.

$$43.897+7.9814x^{2}-13.216x=52.929$$

Subtract 43.897 from both sides.

$$7.9814x^{2}-13.216x=52.929-43.897$$

Subtract 43.897 from 52.929 to get 9.032.

$$7.9814x^{2}-13.216x=9.032$$

Divide both sides of the equation by 7.9814, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{7.9814x^{2}-13.216x}{7.9814}=\frac{9.032}{7.9814}$$

Dividing by 7.9814 undoes the multiplication by 7.9814.

$$x^{2}+\left(-\frac{13.216}{7.9814}\right)x=\frac{9.032}{7.9814}$$

Divide -13.216 by 7.9814 by multiplying -13.216 by the reciprocal of 7.9814.

$$x^{2}-\frac{9440}{5701}x=\frac{9.032}{7.9814}$$

Divide 9.032 by 7.9814 by multiplying 9.032 by the reciprocal of 7.9814.

$$x^{2}-\frac{9440}{5701}x=\frac{45160}{39907}$$

Divide -\frac{9440}{5701}, the coefficient of the x term, by 2 to get -\frac{4720}{5701}. Then add the square of -\frac{4720}{5701} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9440}{5701}x+\left(-\frac{4720}{5701}\right)^{2}=\frac{45160}{39907}+\left(-\frac{4720}{5701}\right)^{2}$$

Square -\frac{4720}{5701} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9440}{5701}x+\frac{22278400}{32501401}=\frac{45160}{39907}+\frac{22278400}{32501401}$$

Add \frac{45160}{39907} to \frac{22278400}{32501401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9440}{5701}x+\frac{22278400}{32501401}=\frac{413405960}{227509807}$$

Factor x^{2}-\frac{9440}{5701}x+\frac{22278400}{32501401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4720}{5701}\right)^{2}=\frac{413405960}{227509807}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4720}{5701}\right)^{2}}=\sqrt{\frac{413405960}{227509807}}$$

Simplify.

$$x-\frac{4720}{5701}=\frac{2\sqrt{723460430}}{39907} x-\frac{4720}{5701}=-\frac{2\sqrt{723460430}}{39907}$$

Add \frac{4720}{5701} to both sides of the equation.

$$x=\frac{2\sqrt{723460430}}{39907}+\frac{4720}{5701} x=-\frac{2\sqrt{723460430}}{39907}+\frac{4720}{5701}$$