Solve -10x^2+210x-800=240

$$-10 { x }^{ 2 } +210x-800=240$$

Solve for x

$$x=8$$

$$x=13$$

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.

$$-10x^{2}+210x-800=240$$

Subtract 240 from both sides of the equation.

$$-10x^{2}+210x-800-240=240-240$$

Subtracting 240 from itself leaves 0.

$$-10x^{2}+210x-800-240=0$$

Subtract 240 from -800.

$$-10x^{2}+210x-1040=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 210 for b, and -1040 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-210±\sqrt{210^{2}-4\left(-10\right)\left(-1040\right)}}{2\left(-10\right)}$$

Square 210.

$$x=\frac{-210±\sqrt{44100-4\left(-10\right)\left(-1040\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-210±\sqrt{44100+40\left(-1040\right)}}{2\left(-10\right)}$$

Multiply 40 times -1040.

$$x=\frac{-210±\sqrt{44100-41600}}{2\left(-10\right)}$$

Add 44100 to -41600.

$$x=\frac{-210±\sqrt{2500}}{2\left(-10\right)}$$

Take the square root of 2500.

$$x=\frac{-210±50}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-210±50}{-20}$$

Now solve the equation x=\frac{-210±50}{-20} when ± is plus. Add -210 to 50.

$$x=-\frac{160}{-20}$$

Divide -160 by -20.

$$x=8$$

Now solve the equation x=\frac{-210±50}{-20} when ± is minus. Subtract 50 from -210.

$$x=-\frac{260}{-20}$$

Divide -260 by -20.

$$x=13$$

The equation is now solved.

$$x=8 x=13$$

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.

$$-10x^{2}+210x-800=240$$

Add 800 to both sides of the equation.

$$-10x^{2}+210x-800-\left(-800\right)=240-\left(-800\right)$$

Subtracting -800 from itself leaves 0.

$$-10x^{2}+210x=240-\left(-800\right)$$

Subtract -800 from 240.

$$-10x^{2}+210x=1040$$

Divide both sides by -10.

$$\frac{-10x^{2}+210x}{-10}=\frac{1040}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{210}{-10}x=\frac{1040}{-10}$$

Divide 210 by -10.

$$x^{2}-21x=\frac{1040}{-10}$$

Divide 1040 by -10.

$$x^{2}-21x=-104$$

Divide -21, the coefficient of the x term, by 2 to get -\frac{21}{2}. Then add the square of -\frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-104+\left(-\frac{21}{2}\right)^{2}$$

Square -\frac{21}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-21x+\frac{441}{4}=-104+\frac{441}{4}$$

Add -104 to \frac{441}{4}.

$$x^{2}-21x+\frac{441}{4}=\frac{25}{4}$$

Factor x^{2}-21x+\frac{441}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{21}{2}=\frac{5}{2} x-\frac{21}{2}=-\frac{5}{2}$$

Add \frac{21}{2} to both sides of the equation.

$$x=13 x=8$$