Solve -5x^2+100x-300=0

$$-5 { x }^{ 2 } +100x-300=0$$

Solve for x

$$x=2\sqrt{10}+10\approx 16.32455532$$

$$x=10-2\sqrt{10}\approx 3.67544468$$

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.

$$-5x^{2}+100x-300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 100 for b, and -300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-5\right)\left(-300\right)}}{2\left(-5\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-5\right)\left(-300\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-100±\sqrt{10000+20\left(-300\right)}}{2\left(-5\right)}$$

Multiply 20 times -300.

$$x=\frac{-100±\sqrt{10000-6000}}{2\left(-5\right)}$$

Add 10000 to -6000.

$$x=\frac{-100±\sqrt{4000}}{2\left(-5\right)}$$

Take the square root of 4000.

$$x=\frac{-100±20\sqrt{10}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-100±20\sqrt{10}}{-10}$$

Now solve the equation x=\frac{-100±20\sqrt{10}}{-10} when ± is plus. Add -100 to 20\sqrt{10}.

$$x=\frac{20\sqrt{10}-100}{-10}$$

Divide -100+20\sqrt{10} by -10.

$$x=10-2\sqrt{10}$$

Now solve the equation x=\frac{-100±20\sqrt{10}}{-10} when ± is minus. Subtract 20\sqrt{10} from -100.

$$x=\frac{-20\sqrt{10}-100}{-10}$$

Divide -100-20\sqrt{10} by -10.

$$x=2\sqrt{10}+10$$

The equation is now solved.

$$x=10-2\sqrt{10} x=2\sqrt{10}+10$$

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.

$$-5x^{2}+100x-300=0$$

Add 300 to both sides of the equation.

$$-5x^{2}+100x-300-\left(-300\right)=-\left(-300\right)$$

Subtracting -300 from itself leaves 0.

$$-5x^{2}+100x=-\left(-300\right)$$

Subtract -300 from 0.

$$-5x^{2}+100x=300$$

Divide both sides by -5.

$$\frac{-5x^{2}+100x}{-5}=\frac{300}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{100}{-5}x=\frac{300}{-5}$$

Divide 100 by -5.

$$x^{2}-20x=\frac{300}{-5}$$

Divide 300 by -5.

$$x^{2}-20x=-60$$

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

$$x^{2}-20x+\left(-10\right)^{2}=-60+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-60+100$$

Add -60 to 100.

$$x^{2}-20x+100=40$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=40$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{40}$$

Simplify.

$$x-10=2\sqrt{10} x-10=-2\sqrt{10}$$

Add 10 to both sides of the equation.

$$x=2\sqrt{10}+10 x=10-2\sqrt{10}$$