Solve 11=1+20x-49x^2

$$11=1+20x-49 { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{390}i+10}{49}\approx 0.204081633-0.403028932i$$

$$x=\frac{10+\sqrt{390}i}{49}\approx 0.204081633+0.403028932i$$

Steps Using the Quadratic Formula

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

$$1+20x-49x^{2}=11$$

Subtract 11 from both sides.

$$1+20x-49x^{2}-11=0$$

Subtract 11 from 1 to get -10.

$$-10+20x-49x^{2}=0$$

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.

$$-49x^{2}+20x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -49 for a, 20 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-49\right)\left(-10\right)}}{2\left(-49\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-49\right)\left(-10\right)}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$x=\frac{-20±\sqrt{400+196\left(-10\right)}}{2\left(-49\right)}$$

Multiply 196 times -10.

$$x=\frac{-20±\sqrt{400-1960}}{2\left(-49\right)}$$

Add 400 to -1960.

$$x=\frac{-20±\sqrt{-1560}}{2\left(-49\right)}$$

Take the square root of -1560.

$$x=\frac{-20±2\sqrt{390}i}{2\left(-49\right)}$$

Multiply 2 times -49.

$$x=\frac{-20±2\sqrt{390}i}{-98}$$

Now solve the equation x=\frac{-20±2\sqrt{390}i}{-98} when ± is plus. Add -20 to 2i\sqrt{390}.

$$x=\frac{-20+2\sqrt{390}i}{-98}$$

Divide -20+2i\sqrt{390} by -98.

$$x=\frac{-\sqrt{390}i+10}{49}$$

Now solve the equation x=\frac{-20±2\sqrt{390}i}{-98} when ± is minus. Subtract 2i\sqrt{390} from -20.

$$x=\frac{-2\sqrt{390}i-20}{-98}$$

Divide -20-2i\sqrt{390} by -98.

$$x=\frac{10+\sqrt{390}i}{49}$$

The equation is now solved.

$$x=\frac{-\sqrt{390}i+10}{49} x=\frac{10+\sqrt{390}i}{49}$$

Steps for Completing the Square

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

$$1+20x-49x^{2}=11$$

Subtract 1 from both sides.

$$20x-49x^{2}=11-1$$

Subtract 1 from 11 to get 10.

$$20x-49x^{2}=10$$

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.

$$-49x^{2}+20x=10$$

Divide both sides by -49.

$$\frac{-49x^{2}+20x}{-49}=\frac{10}{-49}$$

Dividing by -49 undoes the multiplication by -49.

$$x^{2}+\frac{20}{-49}x=\frac{10}{-49}$$

Divide 20 by -49.

$$x^{2}-\frac{20}{49}x=\frac{10}{-49}$$

Divide 10 by -49.

$$x^{2}-\frac{20}{49}x=-\frac{10}{49}$$

Divide -\frac{20}{49}, the coefficient of the x term, by 2 to get -\frac{10}{49}. Then add the square of -\frac{10}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{20}{49}x+\left(-\frac{10}{49}\right)^{2}=-\frac{10}{49}+\left(-\frac{10}{49}\right)^{2}$$

Square -\frac{10}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{20}{49}x+\frac{100}{2401}=-\frac{10}{49}+\frac{100}{2401}$$

Add -\frac{10}{49} to \frac{100}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{20}{49}x+\frac{100}{2401}=-\frac{390}{2401}$$

Factor x^{2}-\frac{20}{49}x+\frac{100}{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}.

$$\left(x-\frac{10}{49}\right)^{2}=-\frac{390}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{10}{49}\right)^{2}}=\sqrt{-\frac{390}{2401}}$$

Simplify.

$$x-\frac{10}{49}=\frac{\sqrt{390}i}{49} x-\frac{10}{49}=-\frac{\sqrt{390}i}{49}$$

Add \frac{10}{49} to both sides of the equation.

$$x=\frac{10+\sqrt{390}i}{49} x=\frac{-\sqrt{390}i+10}{49}$$