Solve 30=-8x-4.9x^2

$$30=-8x-4.9 { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{-40+i\times 10\sqrt{131}}{49}\approx -0.816326531+2.335821049i$$

$$x=\frac{-i\times 10\sqrt{131}-40}{49}\approx -0.816326531-2.335821049i$$

Steps Using the Quadratic Formula

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

$$-8x-4.9x^{2}=30$$

Subtract 30 from both sides.

$$-8x-4.9x^{2}-30=0$$

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.

$$-4.9x^{2}-8x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, -8 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-4.9\right)\left(-30\right)}}{2\left(-4.9\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-4.9\right)\left(-30\right)}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$x=\frac{-\left(-8\right)±\sqrt{64+19.6\left(-30\right)}}{2\left(-4.9\right)}$$

Multiply 19.6 times -30.

$$x=\frac{-\left(-8\right)±\sqrt{64-588}}{2\left(-4.9\right)}$$

Add 64 to -588.

$$x=\frac{-\left(-8\right)±\sqrt{-524}}{2\left(-4.9\right)}$$

Take the square root of -524.

$$x=\frac{-\left(-8\right)±2\sqrt{131}i}{2\left(-4.9\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{131}i}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$x=\frac{8±2\sqrt{131}i}{-9.8}$$

Now solve the equation x=\frac{8±2\sqrt{131}i}{-9.8} when ± is plus. Add 8 to 2i\sqrt{131}.

$$x=\frac{8+2\sqrt{131}i}{-9.8}$$

Divide 8+2i\sqrt{131} by -9.8 by multiplying 8+2i\sqrt{131} by the reciprocal of -9.8.

$$x=\frac{-10\sqrt{131}i-40}{49}$$

Now solve the equation x=\frac{8±2\sqrt{131}i}{-9.8} when ± is minus. Subtract 2i\sqrt{131} from 8.

$$x=\frac{-2\sqrt{131}i+8}{-9.8}$$

Divide 8-2i\sqrt{131} by -9.8 by multiplying 8-2i\sqrt{131} by the reciprocal of -9.8.

$$x=\frac{-40+10\sqrt{131}i}{49}$$

The equation is now solved.

$$x=\frac{-10\sqrt{131}i-40}{49} x=\frac{-40+10\sqrt{131}i}{49}$$

Steps for Completing the Square

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

$$-8x-4.9x^{2}=30$$

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.

$$-4.9x^{2}-8x=30$$

Divide both sides of the equation by -4.9, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-4.9x^{2}-8x}{-4.9}=\frac{30}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$x^{2}+\left(-\frac{8}{-4.9}\right)x=\frac{30}{-4.9}$$

Divide -8 by -4.9 by multiplying -8 by the reciprocal of -4.9.

$$x^{2}+\frac{80}{49}x=\frac{30}{-4.9}$$

Divide 30 by -4.9 by multiplying 30 by the reciprocal of -4.9.

$$x^{2}+\frac{80}{49}x=-\frac{300}{49}$$

Divide \frac{80}{49}, the coefficient of the x term, by 2 to get \frac{40}{49}. Then add the square of \frac{40}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{80}{49}x+\frac{40}{49}^{2}=-\frac{300}{49}+\frac{40}{49}^{2}$$

Square \frac{40}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{80}{49}x+\frac{1600}{2401}=-\frac{300}{49}+\frac{1600}{2401}$$

Add -\frac{300}{49} to \frac{1600}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{80}{49}x+\frac{1600}{2401}=-\frac{13100}{2401}$$

Factor x^{2}+\frac{80}{49}x+\frac{1600}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{40}{49}\right)^{2}=-\frac{13100}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{40}{49}\right)^{2}}=\sqrt{-\frac{13100}{2401}}$$

Simplify.

$$x+\frac{40}{49}=\frac{10\sqrt{131}i}{49} x+\frac{40}{49}=-\frac{10\sqrt{131}i}{49}$$

Subtract \frac{40}{49} from both sides of the equation.

$$x=\frac{-40+10\sqrt{131}i}{49} x=\frac{-10\sqrt{131}i-40}{49}$$