Solve 3m(10-m)=70

$$3 m ( 10 - m ) = 70$$

Solve for m

$$m=\frac{\sqrt{15}}{3}+5\approx 6.290994449$$

$$m=-\frac{\sqrt{15}}{3}+5\approx 3.709005551$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3m by 10-m.

$$30m-3m^{2}=70$$

Subtract 70 from both sides.

$$30m-3m^{2}-70=0$$

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.

$$-3m^{2}+30m-70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 30 for b, and -70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-30±\sqrt{30^{2}-4\left(-3\right)\left(-70\right)}}{2\left(-3\right)}$$

Square 30.

$$m=\frac{-30±\sqrt{900-4\left(-3\right)\left(-70\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$m=\frac{-30±\sqrt{900+12\left(-70\right)}}{2\left(-3\right)}$$

Multiply 12 times -70.

$$m=\frac{-30±\sqrt{900-840}}{2\left(-3\right)}$$

Add 900 to -840.

$$m=\frac{-30±\sqrt{60}}{2\left(-3\right)}$$

Take the square root of 60.

$$m=\frac{-30±2\sqrt{15}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$m=\frac{-30±2\sqrt{15}}{-6}$$

Now solve the equation m=\frac{-30±2\sqrt{15}}{-6} when ± is plus. Add -30 to 2\sqrt{15}.

$$m=\frac{2\sqrt{15}-30}{-6}$$

Divide -30+2\sqrt{15} by -6.

$$m=-\frac{\sqrt{15}}{3}+5$$

Now solve the equation m=\frac{-30±2\sqrt{15}}{-6} when ± is minus. Subtract 2\sqrt{15} from -30.

$$m=\frac{-2\sqrt{15}-30}{-6}$$

Divide -30-2\sqrt{15} by -6.

$$m=\frac{\sqrt{15}}{3}+5$$

The equation is now solved.

$$m=-\frac{\sqrt{15}}{3}+5 m=\frac{\sqrt{15}}{3}+5$$

Steps for Completing the Square

Use the distributive property to multiply 3m by 10-m.

$$30m-3m^{2}=70$$

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.

$$-3m^{2}+30m=70$$

Divide both sides by -3.

$$\frac{-3m^{2}+30m}{-3}=\frac{70}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$m^{2}+\frac{30}{-3}m=\frac{70}{-3}$$

Divide 30 by -3.

$$m^{2}-10m=\frac{70}{-3}$$

Divide 70 by -3.

$$m^{2}-10m=-\frac{70}{3}$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-10m+\left(-5\right)^{2}=-\frac{70}{3}+\left(-5\right)^{2}$$

Square -5.

$$m^{2}-10m+25=-\frac{70}{3}+25$$

Add -\frac{70}{3} to 25.

$$m^{2}-10m+25=\frac{5}{3}$$

Factor m^{2}-10m+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-5\right)^{2}=\frac{5}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-5\right)^{2}}=\sqrt{\frac{5}{3}}$$

Simplify.

$$m-5=\frac{\sqrt{15}}{3} m-5=-\frac{\sqrt{15}}{3}$$

Add 5 to both sides of the equation.

$$m=\frac{\sqrt{15}}{3}+5 m=-\frac{\sqrt{15}}{3}+5$$