Solve 11=1+20x-4.9x^2

$$11=1+20x-4.9 { x }^{ 2 }$$

Solve for x

$$x = \frac{10 \sqrt{51} + 100}{49} \approx 3.4982507$$

$$x=\frac{100-10\sqrt{51}}{49}\approx 0.583381953$$

Steps Using the Quadratic Formula

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

$$1+20x-4.9x^{2}=11$$

Subtract 11 from both sides.

$$1+20x-4.9x^{2}-11=0$$

Subtract 11 from 1 to get -10.

$$-10+20x-4.9x^{2}=0$$

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.

$$-4.9x^{2}+20x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 20 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-4.9\right)\left(-10\right)}}{2\left(-4.9\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-4.9\right)\left(-10\right)}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$x=\frac{-20±\sqrt{400+19.6\left(-10\right)}}{2\left(-4.9\right)}$$

Multiply 19.6 times -10.

$$x=\frac{-20±\sqrt{400-196}}{2\left(-4.9\right)}$$

Add 400 to -196.

$$x=\frac{-20±\sqrt{204}}{2\left(-4.9\right)}$$

Take the square root of 204.

$$x=\frac{-20±2\sqrt{51}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$x=\frac{-20±2\sqrt{51}}{-9.8}$$

Now solve the equation x=\frac{-20±2\sqrt{51}}{-9.8} when ± is plus. Add -20 to 2\sqrt{51}.

$$x=\frac{2\sqrt{51}-20}{-9.8}$$

Divide -20+2\sqrt{51} by -9.8 by multiplying -20+2\sqrt{51} by the reciprocal of -9.8.

$$x=\frac{100-10\sqrt{51}}{49}$$

Now solve the equation x=\frac{-20±2\sqrt{51}}{-9.8} when ± is minus. Subtract 2\sqrt{51} from -20.

$$x=\frac{-2\sqrt{51}-20}{-9.8}$$

Divide -20-2\sqrt{51} by -9.8 by multiplying -20-2\sqrt{51} by the reciprocal of -9.8.

$$x=\frac{10\sqrt{51}+100}{49}$$

The equation is now solved.

$$x=\frac{100-10\sqrt{51}}{49} x=\frac{10\sqrt{51}+100}{49}$$

Steps for Completing the Square

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

$$1+20x-4.9x^{2}=11$$

Subtract 1 from both sides.

$$20x-4.9x^{2}=11-1$$

Subtract 1 from 11 to get 10.

$$20x-4.9x^{2}=10$$

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.

$$-4.9x^{2}+20x=10$$

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

$$\frac{-4.9x^{2}+20x}{-4.9}=\frac{10}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$x^{2}+\frac{20}{-4.9}x=\frac{10}{-4.9}$$

Divide 20 by -4.9 by multiplying 20 by the reciprocal of -4.9.

$$x^{2}-\frac{200}{49}x=\frac{10}{-4.9}$$

Divide 10 by -4.9 by multiplying 10 by the reciprocal of -4.9.

$$x^{2}-\frac{200}{49}x=-\frac{100}{49}$$

Divide -\frac{200}{49}, the coefficient of the x term, by 2 to get -\frac{100}{49}. Then add the square of -\frac{100}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{200}{49}x+\left(-\frac{100}{49}\right)^{2}=-\frac{100}{49}+\left(-\frac{100}{49}\right)^{2}$$

Square -\frac{100}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{200}{49}x+\frac{10000}{2401}=-\frac{100}{49}+\frac{10000}{2401}$$

Add -\frac{100}{49} to \frac{10000}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{200}{49}x+\frac{10000}{2401}=\frac{5100}{2401}$$

Factor x^{2}-\frac{200}{49}x+\frac{10000}{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}.

$$\left(x-\frac{100}{49}\right)^{2}=\frac{5100}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{100}{49}\right)^{2}}=\sqrt{\frac{5100}{2401}}$$

Simplify.

$$x-\frac{100}{49}=\frac{10\sqrt{51}}{49} x-\frac{100}{49}=-\frac{10\sqrt{51}}{49}$$

Add \frac{100}{49} to both sides of the equation.

$$x=\frac{10\sqrt{51}+100}{49} x=\frac{100-10\sqrt{51}}{49}$$