Solve 3+m=-m^2+7m+2/2

$$3+m = \frac{ - { m }^{ 2 } +7m+2 }{ 2 }$$

Solve for m

$$m=1$$

$$m=4$$

Steps Using Factoring

Multiply both sides of the equation by 2.

$$6+2m=-m^{2}+7m+2$$

Add m^{2} to both sides.

$$6+2m+m^{2}=7m+2$$

Subtract 7m from both sides.

$$6+2m+m^{2}-7m=2$$

Combine 2m and -7m to get -5m.

$$6-5m+m^{2}=2$$

Subtract 2 from both sides.

$$6-5m+m^{2}-2=0$$

Subtract 2 from 6 to get 4.

$$4-5m+m^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$m^{2}-5m+4=0$$

To solve the equation, factor m^{2}-5m+4 using formula m^{2}+\left(a+b\right)m+ab=\left(m+a\right)\left(m+b\right). To find a and b, set up a system to be solved.

$$a+b=-5 ab=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -5.

$$a=-4 b=-1$$

Rewrite factored expression \left(m+a\right)\left(m+b\right) using the obtained values.

$$\left(m-4\right)\left(m-1\right)$$

To find equation solutions, solve m-4=0 and m-1=0.

$$m=4 m=1$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 2.

$$6+2m=-m^{2}+7m+2$$

Add m^{2} to both sides.

$$6+2m+m^{2}=7m+2$$

Subtract 7m from both sides.

$$6+2m+m^{2}-7m=2$$

Combine 2m and -7m to get -5m.

$$6-5m+m^{2}=2$$

Subtract 2 from both sides.

$$6-5m+m^{2}-2=0$$

Subtract 2 from 6 to get 4.

$$4-5m+m^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$m^{2}-5m+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as m^{2}+am+bm+4. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\times 4=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -5.

$$a=-4 b=-1$$

Rewrite m^{2}-5m+4 as \left(m^{2}-4m\right)+\left(-m+4\right).

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

Factor out m in the first and -1 in the second group.

$$m\left(m-4\right)-\left(m-4\right)$$

Factor out common term m-4 by using distributive property.

$$\left(m-4\right)\left(m-1\right)$$

To find equation solutions, solve m-4=0 and m-1=0.

$$m=4 m=1$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$6+2m=-m^{2}+7m+2$$

Add m^{2} to both sides.

$$6+2m+m^{2}=7m+2$$

Subtract 7m from both sides.

$$6+2m+m^{2}-7m=2$$

Combine 2m and -7m to get -5m.

$$6-5m+m^{2}=2$$

Subtract 2 from both sides.

$$6-5m+m^{2}-2=0$$

Subtract 2 from 6 to get 4.

$$4-5m+m^{2}=0$$

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.

$$m^{2}-5m+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 4}}{2}$$

Square -5.

$$m=\frac{-\left(-5\right)±\sqrt{25-4\times 4}}{2}$$

Multiply -4 times 4.

$$m=\frac{-\left(-5\right)±\sqrt{25-16}}{2}$$

Add 25 to -16.

$$m=\frac{-\left(-5\right)±\sqrt{9}}{2}$$

Take the square root of 9.

$$m=\frac{-\left(-5\right)±3}{2}$$

The opposite of -5 is 5.

$$m=\frac{5±3}{2}$$

Now solve the equation m=\frac{5±3}{2} when ± is plus. Add 5 to 3.

$$m=\frac{8}{2}$$

Divide 8 by 2.

$$m=4$$

Now solve the equation m=\frac{5±3}{2} when ± is minus. Subtract 3 from 5.

$$m=\frac{2}{2}$$

Divide 2 by 2.

$$m=1$$

The equation is now solved.

$$m=4 m=1$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$6+2m=-m^{2}+7m+2$$

Add m^{2} to both sides.

$$6+2m+m^{2}=7m+2$$

Subtract 7m from both sides.

$$6+2m+m^{2}-7m=2$$

Combine 2m and -7m to get -5m.

$$6-5m+m^{2}=2$$

Subtract 6 from both sides.

$$-5m+m^{2}=2-6$$

Subtract 6 from 2 to get -4.

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

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.

$$m^{2}-5m=-4$$

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

$$m^{2}-5m+\left(-\frac{5}{2}\right)^{2}=-4+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-5m+\frac{25}{4}=-4+\frac{25}{4}$$

Add -4 to \frac{25}{4}.

$$m^{2}-5m+\frac{25}{4}=\frac{9}{4}$$

Factor m^{2}-5m+\frac{25}{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}.

$$\left(m-\frac{5}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$m-\frac{5}{2}=\frac{3}{2} m-\frac{5}{2}=-\frac{3}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$m=4 m=1$$