Solve -0.1x^2+2.2x+5+x-10=20.6

$$- 0.1 x ^ { 2 } + 2.2 x + 5 + x - 10 = 20.6$$

Solve for x

$$x=16$$

Steps Using the Quadratic Formula

Combine 2.2x and x to get 3.2x.

$$-0.1x^{2}+3.2x+5-10=20.6$$

Subtract 10 from 5 to get -5.

$$-0.1x^{2}+3.2x-5=20.6$$

Subtract 20.6 from both sides.

$$-0.1x^{2}+3.2x-5-20.6=0$$

Subtract 20.6 from -5 to get -25.6.

$$-0.1x^{2}+3.2x-25.6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.1 for a, 3.2 for b, and -25.6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3.2±\sqrt{3.2^{2}-4\left(-0.1\right)\left(-25.6\right)}}{2\left(-0.1\right)}$$

Square 3.2 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-3.2±\sqrt{10.24-4\left(-0.1\right)\left(-25.6\right)}}{2\left(-0.1\right)}$$

Multiply -4 times -0.1.

$$x=\frac{-3.2±\sqrt{10.24+0.4\left(-25.6\right)}}{2\left(-0.1\right)}$$

Multiply 0.4 times -25.6 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-3.2±\sqrt{\frac{256-256}{25}}}{2\left(-0.1\right)}$$

Add 10.24 to -10.24 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-3.2±\sqrt{0}}{2\left(-0.1\right)}$$

Take the square root of 0.

$$x=-\frac{3.2}{2\left(-0.1\right)}$$

Multiply 2 times -0.1.

$$x=-\frac{3.2}{-0.2}$$

Divide -3.2 by -0.2 by multiplying -3.2 by the reciprocal of -0.2.

$$x=16$$

Steps for Completing the Square

Combine 2.2x and x to get 3.2x.

$$-0.1x^{2}+3.2x+5-10=20.6$$

Subtract 10 from 5 to get -5.

$$-0.1x^{2}+3.2x-5=20.6$$

Add 5 to both sides.

$$-0.1x^{2}+3.2x=20.6+5$$

Add 20.6 and 5 to get 25.6.

$$-0.1x^{2}+3.2x=25.6$$

Multiply both sides by -10.

$$\frac{-0.1x^{2}+3.2x}{-0.1}=\frac{25.6}{-0.1}$$

Dividing by -0.1 undoes the multiplication by -0.1.

$$x^{2}+\frac{3.2}{-0.1}x=\frac{25.6}{-0.1}$$

Divide 3.2 by -0.1 by multiplying 3.2 by the reciprocal of -0.1.

$$x^{2}-32x=\frac{25.6}{-0.1}$$

Divide 25.6 by -0.1 by multiplying 25.6 by the reciprocal of -0.1.

$$x^{2}-32x=-256$$

Divide -32, the coefficient of the x term, by 2 to get -16. Then add the square of -16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-32x+\left(-16\right)^{2}=-256+\left(-16\right)^{2}$$

Square -16.

$$x^{2}-32x+256=-256+256$$

Add -256 to 256.

$$x^{2}-32x+256=0$$

Factor x^{2}-32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-16\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-16\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-16=0 x-16=0$$

Add 16 to both sides of the equation.

$$x=16 x=16$$

The equation is now solved. Solutions are the same.

$$x=16$$