Solve 30=8x+0.5*9.8x^2

$$30=8x+0.5 \times 9.8 { x }^{ 2 }$$

Solve for x

$$x = \frac{10 \sqrt{163} - 40}{49} \approx 1.789213334$$

$$x=\frac{-10\sqrt{163}-40}{49}\approx -3.421866395$$

Steps Using the Quadratic Formula

Multiply 0.5 and 9.8 to get 4.9.

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

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}.

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

Square 8.

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

Multiply -4 times 4.9.

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

Multiply -19.6 times -30.

$$x=\frac{-8±\sqrt{64+588}}{2\times 4.9}$$

Add 64 to 588.

$$x=\frac{-8±\sqrt{652}}{2\times 4.9}$$

Take the square root of 652.

$$x=\frac{-8±2\sqrt{163}}{2\times 4.9}$$

Multiply 2 times 4.9.

$$x=\frac{-8±2\sqrt{163}}{9.8}$$

Now solve the equation x=\frac{-8±2\sqrt{163}}{9.8} when ± is plus. Add -8 to 2\sqrt{163}.

$$x=\frac{2\sqrt{163}-8}{9.8}$$

Divide -8+2\sqrt{163} by 9.8 by multiplying -8+2\sqrt{163} by the reciprocal of 9.8.

$$x=\frac{10\sqrt{163}-40}{49}$$

Now solve the equation x=\frac{-8±2\sqrt{163}}{9.8} when ± is minus. Subtract 2\sqrt{163} from -8.

$$x=\frac{-2\sqrt{163}-8}{9.8}$$

Divide -8-2\sqrt{163} by 9.8 by multiplying -8-2\sqrt{163} by the reciprocal of 9.8.

$$x=\frac{-10\sqrt{163}-40}{49}$$

The equation is now solved.

$$x=\frac{10\sqrt{163}-40}{49} x=\frac{-10\sqrt{163}-40}{49}$$

Steps for Completing the Square

Multiply 0.5 and 9.8 to get 4.9.

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

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.

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

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.

$$x^{2}+\frac{80}{49}x+\frac{1600}{2401}=\frac{16300}{2401}$$

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}.

$$\left(x+\frac{40}{49}\right)^{2}=\frac{16300}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{40}{49}\right)^{2}}=\sqrt{\frac{16300}{2401}}$$

Simplify.

$$x+\frac{40}{49}=\frac{10\sqrt{163}}{49} x+\frac{40}{49}=-\frac{10\sqrt{163}}{49}$$

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

$$x=\frac{10\sqrt{163}-40}{49} x=\frac{-10\sqrt{163}-40}{49}$$