Solve 3b+1+b^2/4-2b=16

$$3 b + 1 + \frac { b ^ { 2 } } { 4 } - 2 b = 16$$

Solve for b

$$b=-10$$

$$b=6$$

Steps Using Factoring

Multiply both sides of the equation by 4.

$$12b+4+b^{2}-8b=64$$

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

$$4b+4+b^{2}=64$$

Subtract 64 from both sides.

$$4b+4+b^{2}-64=0$$

Subtract 64 from 4 to get -60.

$$4b-60+b^{2}=0$$

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

$$b^{2}+4b-60=0$$

To solve the equation, factor b^{2}+4b-60 using formula b^{2}+\left(a+b\right)b+ab=\left(b+a\right)\left(b+b\right). To find a and b, set up a system to be solved.

$$a+b=4 ab=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 4.

$$a=-6 b=10$$

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

$$\left(b-6\right)\left(b+10\right)$$

To find equation solutions, solve b-6=0 and b+10=0.

$$b=6 b=-10$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 4.

$$12b+4+b^{2}-8b=64$$

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

$$4b+4+b^{2}=64$$

Subtract 64 from both sides.

$$4b+4+b^{2}-64=0$$

Subtract 64 from 4 to get -60.

$$4b-60+b^{2}=0$$

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

$$b^{2}+4b-60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as b^{2}+ab+bb-60. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\left(-60\right)=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 4.

$$a=-6 b=10$$

Rewrite b^{2}+4b-60 as \left(b^{2}-6b\right)+\left(10b-60\right).

$$\left(b^{2}-6b\right)+\left(10b-60\right)$$

Factor out b in the first and 10 in the second group.

$$b\left(b-6\right)+10\left(b-6\right)$$

Factor out common term b-6 by using distributive property.

$$\left(b-6\right)\left(b+10\right)$$

To find equation solutions, solve b-6=0 and b+10=0.

$$b=6 b=-10$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4.

$$12b+4+b^{2}-8b=64$$

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

$$4b+4+b^{2}=64$$

Subtract 64 from both sides.

$$4b+4+b^{2}-64=0$$

Subtract 64 from 4 to get -60.

$$4b-60+b^{2}=0$$

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.

$$b^{2}+4b-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-4±\sqrt{4^{2}-4\left(-60\right)}}{2}$$

Square 4.

$$b=\frac{-4±\sqrt{16-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$b=\frac{-4±\sqrt{16+240}}{2}$$

Add 16 to 240.

$$b=\frac{-4±\sqrt{256}}{2}$$

Take the square root of 256.

$$b=\frac{-4±16}{2}$$

Now solve the equation b=\frac{-4±16}{2} when ± is plus. Add -4 to 16.

$$b=\frac{12}{2}$$

Divide 12 by 2.

$$b=6$$

Now solve the equation b=\frac{-4±16}{2} when ± is minus. Subtract 16 from -4.

$$b=-\frac{20}{2}$$

Divide -20 by 2.

$$b=-10$$

The equation is now solved.

$$b=6 b=-10$$

Steps for Completing the Square

Multiply both sides of the equation by 4.

$$12b+4+b^{2}-8b=64$$

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

$$4b+4+b^{2}=64$$

Subtract 4 from both sides.

$$4b+b^{2}=64-4$$

Subtract 4 from 64 to get 60.

$$4b+b^{2}=60$$

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.

$$b^{2}+4b=60$$

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.

$$b^{2}+4b+2^{2}=60+2^{2}$$

Square 2.

$$b^{2}+4b+4=60+4$$

Add 60 to 4.

$$b^{2}+4b+4=64$$

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

$$\left(b+2\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b+2\right)^{2}}=\sqrt{64}$$

Simplify.

$$b+2=8 b+2=-8$$

Subtract 2 from both sides of the equation.

$$b=6 b=-10$$