Solve 30=25x+1/2-9.81x^2

$$30 = 25x+ \frac{ 1 }{ 2 } -9.81 { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{1250+i\times 5\sqrt{53258}}{981}\approx 1.27420999+1.176233173i$$

$$x=\frac{-i\times 5\sqrt{53258}+1250}{981}\approx 1.27420999-1.176233173i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$25x+\frac{1}{2}-9.81x^{2}=30$$

Subtract 30 from both sides.

$$25x+\frac{1}{2}-9.81x^{2}-30=0$$

Subtract 30 from \frac{1}{2} to get -\frac{59}{2}.

$$25x-\frac{59}{2}-9.81x^{2}=0$$

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.

$$-9.81x^{2}+25x-\frac{59}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9.81 for a, 25 for b, and -\frac{59}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-9.81\right)\left(-\frac{59}{2}\right)}}{2\left(-9.81\right)}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\left(-9.81\right)\left(-\frac{59}{2}\right)}}{2\left(-9.81\right)}$$

Multiply -4 times -9.81.

$$x=\frac{-25±\sqrt{625+39.24\left(-\frac{59}{2}\right)}}{2\left(-9.81\right)}$$

Multiply 39.24 times -\frac{59}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-25±\sqrt{625-\frac{57879}{50}}}{2\left(-9.81\right)}$$

Add 625 to -\frac{57879}{50}.

$$x=\frac{-25±\sqrt{-\frac{26629}{50}}}{2\left(-9.81\right)}$$

Take the square root of -\frac{26629}{50}.

$$x=\frac{-25±\frac{\sqrt{53258}i}{10}}{2\left(-9.81\right)}$$

Multiply 2 times -9.81.

$$x=\frac{-25±\frac{\sqrt{53258}i}{10}}{-19.62}$$

Now solve the equation x=\frac{-25±\frac{\sqrt{53258}i}{10}}{-19.62} when ± is plus. Add -25 to \frac{i\sqrt{53258}}{10}.

$$x=\frac{\frac{\sqrt{53258}i}{10}-25}{-19.62}$$

Divide -25+\frac{i\sqrt{53258}}{10} by -19.62 by multiplying -25+\frac{i\sqrt{53258}}{10} by the reciprocal of -19.62.

$$x=\frac{-5\sqrt{53258}i+1250}{981}$$

Now solve the equation x=\frac{-25±\frac{\sqrt{53258}i}{10}}{-19.62} when ± is minus. Subtract \frac{i\sqrt{53258}}{10} from -25.

$$x=\frac{-\frac{\sqrt{53258}i}{10}-25}{-19.62}$$

Divide -25-\frac{i\sqrt{53258}}{10} by -19.62 by multiplying -25-\frac{i\sqrt{53258}}{10} by the reciprocal of -19.62.

$$x=\frac{1250+5\sqrt{53258}i}{981}$$

The equation is now solved.

$$x=\frac{-5\sqrt{53258}i+1250}{981} x=\frac{1250+5\sqrt{53258}i}{981}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$25x+\frac{1}{2}-9.81x^{2}=30$$

Subtract \frac{1}{2} from both sides.

$$25x-9.81x^{2}=30-\frac{1}{2}$$

Subtract \frac{1}{2} from 30 to get \frac{59}{2}.

$$25x-9.81x^{2}=\frac{59}{2}$$

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.

$$-9.81x^{2}+25x=\frac{59}{2}$$

Divide both sides of the equation by -9.81, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-9.81x^{2}+25x}{-9.81}=\frac{\frac{59}{2}}{-9.81}$$

Dividing by -9.81 undoes the multiplication by -9.81.

$$x^{2}+\frac{25}{-9.81}x=\frac{\frac{59}{2}}{-9.81}$$

Divide 25 by -9.81 by multiplying 25 by the reciprocal of -9.81.

$$x^{2}-\frac{2500}{981}x=\frac{\frac{59}{2}}{-9.81}$$

Divide \frac{59}{2} by -9.81 by multiplying \frac{59}{2} by the reciprocal of -9.81.

$$x^{2}-\frac{2500}{981}x=-\frac{2950}{981}$$

Divide -\frac{2500}{981}, the coefficient of the x term, by 2 to get -\frac{1250}{981}. Then add the square of -\frac{1250}{981} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2500}{981}x+\left(-\frac{1250}{981}\right)^{2}=-\frac{2950}{981}+\left(-\frac{1250}{981}\right)^{2}$$

Square -\frac{1250}{981} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2500}{981}x+\frac{1562500}{962361}=-\frac{2950}{981}+\frac{1562500}{962361}$$

Add -\frac{2950}{981} to \frac{1562500}{962361} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2500}{981}x+\frac{1562500}{962361}=-\frac{1331450}{962361}$$

Factor x^{2}-\frac{2500}{981}x+\frac{1562500}{962361}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1250}{981}\right)^{2}=-\frac{1331450}{962361}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1250}{981}\right)^{2}}=\sqrt{-\frac{1331450}{962361}}$$

Simplify.

$$x-\frac{1250}{981}=\frac{5\sqrt{53258}i}{981} x-\frac{1250}{981}=-\frac{5\sqrt{53258}i}{981}$$

Add \frac{1250}{981} to both sides of the equation.

$$x=\frac{1250+5\sqrt{53258}i}{981} x=\frac{-5\sqrt{53258}i+1250}{981}$$