Solve 10m^2+35m^2+18m-126=0

$$10 m ^ { 2 } + 35 m ^ { 2 } + 18 m - 126 = 0$$

Solve for m

$$m = \frac{\sqrt{71} - 1}{5} \approx 1.485229955$$

$$m=\frac{-\sqrt{71}-1}{5}\approx -1.885229955$$

Steps Using the Quadratic Formula

Combine 10m^{2} and 35m^{2} to get 45m^{2}.

$$45m^{2}+18m-126=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 45 for a, 18 for b, and -126 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-18±\sqrt{18^{2}-4\times 45\left(-126\right)}}{2\times 45}$$

Square 18.

$$m=\frac{-18±\sqrt{324-4\times 45\left(-126\right)}}{2\times 45}$$

Multiply -4 times 45.

$$m=\frac{-18±\sqrt{324-180\left(-126\right)}}{2\times 45}$$

Multiply -180 times -126.

$$m=\frac{-18±\sqrt{324+22680}}{2\times 45}$$

Add 324 to 22680.

$$m=\frac{-18±\sqrt{23004}}{2\times 45}$$

Take the square root of 23004.

$$m=\frac{-18±18\sqrt{71}}{2\times 45}$$

Multiply 2 times 45.

$$m=\frac{-18±18\sqrt{71}}{90}$$

Now solve the equation m=\frac{-18±18\sqrt{71}}{90} when ± is plus. Add -18 to 18\sqrt{71}.

$$m=\frac{18\sqrt{71}-18}{90}$$

Divide -18+18\sqrt{71} by 90.

$$m=\frac{\sqrt{71}-1}{5}$$

Now solve the equation m=\frac{-18±18\sqrt{71}}{90} when ± is minus. Subtract 18\sqrt{71} from -18.

$$m=\frac{-18\sqrt{71}-18}{90}$$

Divide -18-18\sqrt{71} by 90.

$$m=\frac{-\sqrt{71}-1}{5}$$

The equation is now solved.

$$m=\frac{\sqrt{71}-1}{5} m=\frac{-\sqrt{71}-1}{5}$$

Steps for Completing the Square

Combine 10m^{2} and 35m^{2} to get 45m^{2}.

$$45m^{2}+18m-126=0$$

Add 126 to both sides. Anything plus zero gives itself.

$$45m^{2}+18m=126$$

Divide both sides by 45.

$$\frac{45m^{2}+18m}{45}=\frac{126}{45}$$

Dividing by 45 undoes the multiplication by 45.

$$m^{2}+\frac{18}{45}m=\frac{126}{45}$$

Reduce the fraction \frac{18}{45} to lowest terms by extracting and canceling out 9.

$$m^{2}+\frac{2}{5}m=\frac{126}{45}$$

Reduce the fraction \frac{126}{45} to lowest terms by extracting and canceling out 9.

$$m^{2}+\frac{2}{5}m=\frac{14}{5}$$

Divide \frac{2}{5}, the coefficient of the x term, by 2 to get \frac{1}{5}. Then add the square of \frac{1}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{2}{5}m+\left(\frac{1}{5}\right)^{2}=\frac{14}{5}+\left(\frac{1}{5}\right)^{2}$$

Square \frac{1}{5} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{2}{5}m+\frac{1}{25}=\frac{14}{5}+\frac{1}{25}$$

Add \frac{14}{5} to \frac{1}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{2}{5}m+\frac{1}{25}=\frac{71}{25}$$

Factor m^{2}+\frac{2}{5}m+\frac{1}{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}.

$$\left(m+\frac{1}{5}\right)^{2}=\frac{71}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{1}{5}\right)^{2}}=\sqrt{\frac{71}{25}}$$

Simplify.

$$m+\frac{1}{5}=\frac{\sqrt{71}}{5} m+\frac{1}{5}=-\frac{\sqrt{71}}{5}$$

Subtract \frac{1}{5} from both sides of the equation.

$$m=\frac{\sqrt{71}-1}{5} m=\frac{-\sqrt{71}-1}{5}$$