Solve -0.1x^2+10x-210=0

$$-0.1 { x }^{ 2 } +10x-210=0$$

Solve for x

$$x=30$$

$$x=70$$

Steps Using the Quadratic Formula

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.

$$-0.1x^{2}+10x-210=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.1 for a, 10 for b, and -210 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-0.1\right)\left(-210\right)}}{2\left(-0.1\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-0.1\right)\left(-210\right)}}{2\left(-0.1\right)}$$

Multiply -4 times -0.1.

$$x=\frac{-10±\sqrt{100+0.4\left(-210\right)}}{2\left(-0.1\right)}$$

Multiply 0.4 times -210.

$$x=\frac{-10±\sqrt{100-84}}{2\left(-0.1\right)}$$

Add 100 to -84.

$$x=\frac{-10±\sqrt{16}}{2\left(-0.1\right)}$$

Take the square root of 16.

$$x=\frac{-10±4}{2\left(-0.1\right)}$$

Multiply 2 times -0.1.

$$x=\frac{-10±4}{-0.2}$$

Now solve the equation x=\frac{-10±4}{-0.2} when ± is plus. Add -10 to 4.

$$x=-\frac{6}{-0.2}$$

Divide -6 by -0.2 by multiplying -6 by the reciprocal of -0.2.

$$x=30$$

Now solve the equation x=\frac{-10±4}{-0.2} when ± is minus. Subtract 4 from -10.

$$x=-\frac{14}{-0.2}$$

Divide -14 by -0.2 by multiplying -14 by the reciprocal of -0.2.

$$x=70$$

The equation is now solved.

$$x=30 x=70$$

Steps for Completing the Square

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.

$$-0.1x^{2}+10x-210=0$$

Add 210 to both sides of the equation.

$$-0.1x^{2}+10x-210-\left(-210\right)=-\left(-210\right)$$

Subtracting -210 from itself leaves 0.

$$-0.1x^{2}+10x=-\left(-210\right)$$

Subtract -210 from 0.

$$-0.1x^{2}+10x=210$$

Multiply both sides by -10.

$$\frac{-0.1x^{2}+10x}{-0.1}=\frac{210}{-0.1}$$

Dividing by -0.1 undoes the multiplication by -0.1.

$$x^{2}+\frac{10}{-0.1}x=\frac{210}{-0.1}$$

Divide 10 by -0.1 by multiplying 10 by the reciprocal of -0.1.

$$x^{2}-100x=\frac{210}{-0.1}$$

Divide 210 by -0.1 by multiplying 210 by the reciprocal of -0.1.

$$x^{2}-100x=-2100$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-2100+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-2100+2500$$

Add -2100 to 2500.

$$x^{2}-100x+2500=400$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{400}$$

Simplify.

$$x-50=20 x-50=-20$$

Add 50 to both sides of the equation.

$$x=70 x=30$$