Solve 12=(14-b)*b/2/2

$$12 = \frac { ( 14 - b ) \cdot \frac { b } { 2 } } { 2 }$$

Solve for b

$$b=6$$

$$b=8$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$12\times 2=\left(14-b\right)\times \frac{b}{2}$$

Multiply both sides of the equation by 2.

$$24\times 2=\left(14-b\right)b$$

Multiply 24 and 2 to get 48.

$$48=\left(14-b\right)b$$

Use the distributive property to multiply 14-b by b.

$$48=14b-b^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$14b-b^{2}=48$$

Subtract 48 from both sides.

$$14b-b^{2}-48=0$$

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.

$$-b^{2}+14b-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 14 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\left(-48\right)}}{2\left(-1\right)}$$

Square 14.

$$b=\frac{-14±\sqrt{196-4\left(-1\right)\left(-48\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$b=\frac{-14±\sqrt{196+4\left(-48\right)}}{2\left(-1\right)}$$

Multiply 4 times -48.

$$b=\frac{-14±\sqrt{196-192}}{2\left(-1\right)}$$

Add 196 to -192.

$$b=\frac{-14±\sqrt{4}}{2\left(-1\right)}$$

Take the square root of 4.

$$b=\frac{-14±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$b=\frac{-14±2}{-2}$$

Now solve the equation b=\frac{-14±2}{-2} when ± is plus. Add -14 to 2.

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

Divide -12 by -2.

$$b=6$$

Now solve the equation b=\frac{-14±2}{-2} when ± is minus. Subtract 2 from -14.

$$b=-\frac{16}{-2}$$

Divide -16 by -2.

$$b=8$$

The equation is now solved.

$$b=6 b=8$$

Steps for Completing the Square

Multiply both sides by 2.

$$12\times 2=\left(14-b\right)\times \frac{b}{2}$$

Multiply both sides of the equation by 2.

$$24\times 2=\left(14-b\right)b$$

Multiply 24 and 2 to get 48.

$$48=\left(14-b\right)b$$

Use the distributive property to multiply 14-b by b.

$$48=14b-b^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$14b-b^{2}=48$$

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.

$$-b^{2}+14b=48$$

Divide both sides by -1.

$$\frac{-b^{2}+14b}{-1}=\frac{48}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$b^{2}+\frac{14}{-1}b=\frac{48}{-1}$$

Divide 14 by -1.

$$b^{2}-14b=\frac{48}{-1}$$

Divide 48 by -1.

$$b^{2}-14b=-48$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-14b+\left(-7\right)^{2}=-48+\left(-7\right)^{2}$$

Square -7.

$$b^{2}-14b+49=-48+49$$

Add -48 to 49.

$$b^{2}-14b+49=1$$

Factor b^{2}-14b+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-7\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-7\right)^{2}}=\sqrt{1}$$

Simplify.

$$b-7=1 b-7=-1$$

Add 7 to both sides of the equation.

$$b=8 b=6$$