Solve -5x^2+700x+28000=0

$$-5 { x }^{ 2 } +700x+28000 = 0$$

Solve for x

$$x=10\sqrt{105}+70\approx 172.46950766$$

$$x=70-10\sqrt{105}\approx -32.46950766$$

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.

$$-5x^{2}+700x+28000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 700 for b, and 28000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-700±\sqrt{700^{2}-4\left(-5\right)\times 28000}}{2\left(-5\right)}$$

Square 700.

$$x=\frac{-700±\sqrt{490000-4\left(-5\right)\times 28000}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-700±\sqrt{490000+20\times 28000}}{2\left(-5\right)}$$

Multiply 20 times 28000.

$$x=\frac{-700±\sqrt{490000+560000}}{2\left(-5\right)}$$

Add 490000 to 560000.

$$x=\frac{-700±\sqrt{1050000}}{2\left(-5\right)}$$

Take the square root of 1050000.

$$x=\frac{-700±100\sqrt{105}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-700±100\sqrt{105}}{-10}$$

Now solve the equation x=\frac{-700±100\sqrt{105}}{-10} when ± is plus. Add -700 to 100\sqrt{105}.

$$x=\frac{100\sqrt{105}-700}{-10}$$

Divide -700+100\sqrt{105} by -10.

$$x=70-10\sqrt{105}$$

Now solve the equation x=\frac{-700±100\sqrt{105}}{-10} when ± is minus. Subtract 100\sqrt{105} from -700.

$$x=\frac{-100\sqrt{105}-700}{-10}$$

Divide -700-100\sqrt{105} by -10.

$$x=10\sqrt{105}+70$$

The equation is now solved.

$$x=70-10\sqrt{105} x=10\sqrt{105}+70$$

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.

$$-5x^{2}+700x+28000=0$$

Subtract 28000 from both sides of the equation.

$$-5x^{2}+700x+28000-28000=-28000$$

Subtracting 28000 from itself leaves 0.

$$-5x^{2}+700x=-28000$$

Divide both sides by -5.

$$\frac{-5x^{2}+700x}{-5}=-\frac{28000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{700}{-5}x=-\frac{28000}{-5}$$

Divide 700 by -5.

$$x^{2}-140x=-\frac{28000}{-5}$$

Divide -28000 by -5.

$$x^{2}-140x=5600$$

Divide -140, the coefficient of the x term, by 2 to get -70. Then add the square of -70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-140x+\left(-70\right)^{2}=5600+\left(-70\right)^{2}$$

Square -70.

$$x^{2}-140x+4900=5600+4900$$

Add 5600 to 4900.

$$x^{2}-140x+4900=10500$$

Factor x^{2}-140x+4900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-70\right)^{2}=10500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-70\right)^{2}}=\sqrt{10500}$$

Simplify.

$$x-70=10\sqrt{105} x-70=-10\sqrt{105}$$

Add 70 to both sides of the equation.

$$x=10\sqrt{105}+70 x=70-10\sqrt{105}$$