Solve left(m-4right)^2-4mleft(m+1right)=0

$${ \left(m-4 \right) }^{ 2 } -4m \left( m+1 \right) =0$$

Solve for m

$$m=\frac{2\sqrt{21}}{3}-2\approx 1.055050463$$

$$m=-\frac{2\sqrt{21}}{3}-2\approx -5.055050463$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-4\right)^{2}.

$$m^{2}-8m+16-4m\left(m+1\right)=0$$

Use the distributive property to multiply -4m by m+1.

$$m^{2}-8m+16-4m^{2}-4m=0$$

Combine m^{2} and -4m^{2} to get -3m^{2}.

$$-3m^{2}-8m+16-4m=0$$

Combine -8m and -4m to get -12m.

$$-3m^{2}-12m+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -12 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 16}}{2\left(-3\right)}$$

Square -12.

$$m=\frac{-\left(-12\right)±\sqrt{144-4\left(-3\right)\times 16}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$m=\frac{-\left(-12\right)±\sqrt{144+12\times 16}}{2\left(-3\right)}$$

Multiply 12 times 16.

$$m=\frac{-\left(-12\right)±\sqrt{144+192}}{2\left(-3\right)}$$

Add 144 to 192.

$$m=\frac{-\left(-12\right)±\sqrt{336}}{2\left(-3\right)}$$

Take the square root of 336.

$$m=\frac{-\left(-12\right)±4\sqrt{21}}{2\left(-3\right)}$$

The opposite of -12 is 12.

$$m=\frac{12±4\sqrt{21}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$m=\frac{12±4\sqrt{21}}{-6}$$

Now solve the equation m=\frac{12±4\sqrt{21}}{-6} when ± is plus. Add 12 to 4\sqrt{21}.

$$m=\frac{4\sqrt{21}+12}{-6}$$

Divide 12+4\sqrt{21} by -6.

$$m=-\frac{2\sqrt{21}}{3}-2$$

Now solve the equation m=\frac{12±4\sqrt{21}}{-6} when ± is minus. Subtract 4\sqrt{21} from 12.

$$m=\frac{12-4\sqrt{21}}{-6}$$

Divide 12-4\sqrt{21} by -6.

$$m=\frac{2\sqrt{21}}{3}-2$$

The equation is now solved.

$$m=-\frac{2\sqrt{21}}{3}-2 m=\frac{2\sqrt{21}}{3}-2$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-4\right)^{2}.

$$m^{2}-8m+16-4m\left(m+1\right)=0$$

Use the distributive property to multiply -4m by m+1.

$$m^{2}-8m+16-4m^{2}-4m=0$$

Combine m^{2} and -4m^{2} to get -3m^{2}.

$$-3m^{2}-8m+16-4m=0$$

Combine -8m and -4m to get -12m.

$$-3m^{2}-12m+16=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$-3m^{2}-12m=-16$$

Divide both sides by -3.

$$\frac{-3m^{2}-12m}{-3}=-\frac{16}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$m^{2}+\left(-\frac{12}{-3}\right)m=-\frac{16}{-3}$$

Divide -12 by -3.

$$m^{2}+4m=-\frac{16}{-3}$$

Divide -16 by -3.

$$m^{2}+4m=\frac{16}{3}$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+4m+2^{2}=\frac{16}{3}+2^{2}$$

Square 2.

$$m^{2}+4m+4=\frac{16}{3}+4$$

Add \frac{16}{3} to 4.

$$m^{2}+4m+4=\frac{28}{3}$$

Factor m^{2}+4m+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}.

$$\left(m+2\right)^{2}=\frac{28}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+2\right)^{2}}=\sqrt{\frac{28}{3}}$$

Simplify.

$$m+2=\frac{2\sqrt{21}}{3} m+2=-\frac{2\sqrt{21}}{3}$$

Subtract 2 from both sides of the equation.

$$m=\frac{2\sqrt{21}}{3}-2 m=-\frac{2\sqrt{21}}{3}-2$$