Solve -m-1-m^2-3m+4=5

$$- m - 1 - m ^ { 2 } - 3 m + 4 = 5$$

Solve for m

$$m=\sqrt{2}-2\approx -0.585786438$$

$$m=-\sqrt{2}-2\approx -3.414213562$$

Steps Using the Quadratic Formula

Add -1 and 4 to get 3.

$$-m+3-m^{2}-3m=5$$

Subtract 5 from both sides.

$$-m+3-m^{2}-3m-5=0$$

Subtract 5 from 3 to get -2.

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

Combine -m and -3m to get -4m.

$$-4m-2-m^{2}=0$$

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.

$$-m^{2}-4m-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Square -4.

$$m=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$m=\frac{-\left(-4\right)±\sqrt{16+4\left(-2\right)}}{2\left(-1\right)}$$

Multiply 4 times -2.

$$m=\frac{-\left(-4\right)±\sqrt{16-8}}{2\left(-1\right)}$$

Add 16 to -8.

$$m=\frac{-\left(-4\right)±\sqrt{8}}{2\left(-1\right)}$$

Take the square root of 8.

$$m=\frac{-\left(-4\right)±2\sqrt{2}}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$m=\frac{4±2\sqrt{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{4±2\sqrt{2}}{-2}$$

Now solve the equation m=\frac{4±2\sqrt{2}}{-2} when ± is plus. Add 4 to 2\sqrt{2}.

$$m=\frac{2\sqrt{2}+4}{-2}$$

Divide 4+2\sqrt{2} by -2.

$$m=-\sqrt{2}-2$$

Now solve the equation m=\frac{4±2\sqrt{2}}{-2} when ± is minus. Subtract 2\sqrt{2} from 4.

$$m=\frac{4-2\sqrt{2}}{-2}$$

Divide 4-2\sqrt{2} by -2.

$$m=\sqrt{2}-2$$

The equation is now solved.

$$m=-\sqrt{2}-2 m=\sqrt{2}-2$$

Steps for Completing the Square

Add -1 and 4 to get 3.

$$-m+3-m^{2}-3m=5$$

Subtract 3 from both sides.

$$-m-m^{2}-3m=5-3$$

Subtract 3 from 5 to get 2.

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

Combine -m and -3m to get -4m.

$$-4m-m^{2}=2$$

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.

$$-m^{2}-4m=2$$

Divide both sides by -1.

$$\frac{-m^{2}-4m}{-1}=\frac{2}{-1}$$

Dividing by -1 undoes the multiplication by -1.

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

Divide -4 by -1.

$$m^{2}+4m=\frac{2}{-1}$$

Divide 2 by -1.

$$m^{2}+4m=-2$$

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.

$$m^{2}+4m+2^{2}=-2+2^{2}$$

Square 2.

$$m^{2}+4m+4=-2+4$$

Add -2 to 4.

$$m^{2}+4m+4=2$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$m+2=\sqrt{2} m+2=-\sqrt{2}$$

Subtract 2 from both sides of the equation.

$$m=\sqrt{2}-2 m=-\sqrt{2}-2$$