Solve b^2+12(5-b)=0

$$b ^ { 2 } + 12 ( 5 - b ) = 0$$

Solve for b

$$b=6+2\sqrt{6}i\approx 6+4.898979486i$$

$$b=-2\sqrt{6}i+6\approx 6-4.898979486i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12 by 5-b.

$$b^{2}+60-12b=0$$

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.

$$b^{2}-12b+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 60}}{2}$$

Square -12.

$$b=\frac{-\left(-12\right)±\sqrt{144-4\times 60}}{2}$$

Multiply -4 times 60.

$$b=\frac{-\left(-12\right)±\sqrt{144-240}}{2}$$

Add 144 to -240.

$$b=\frac{-\left(-12\right)±\sqrt{-96}}{2}$$

Take the square root of -96.

$$b=\frac{-\left(-12\right)±4\sqrt{6}i}{2}$$

The opposite of -12 is 12.

$$b=\frac{12±4\sqrt{6}i}{2}$$

Now solve the equation b=\frac{12±4\sqrt{6}i}{2} when ± is plus. Add 12 to 4i\sqrt{6}.

$$b=\frac{12+4\sqrt{6}i}{2}$$

Divide 12+4i\sqrt{6} by 2.

$$b=6+2\sqrt{6}i$$

Now solve the equation b=\frac{12±4\sqrt{6}i}{2} when ± is minus. Subtract 4i\sqrt{6} from 12.

$$b=\frac{-4\sqrt{6}i+12}{2}$$

Divide 12-4i\sqrt{6} by 2.

$$b=-2\sqrt{6}i+6$$

The equation is now solved.

$$b=6+2\sqrt{6}i b=-2\sqrt{6}i+6$$

Steps for Completing the Square

Use the distributive property to multiply 12 by 5-b.

$$b^{2}+60-12b=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$b^{2}-12b=-60$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-12b+\left(-6\right)^{2}=-60+\left(-6\right)^{2}$$

Square -6.

$$b^{2}-12b+36=-60+36$$

Add -60 to 36.

$$b^{2}-12b+36=-24$$

Factor b^{2}-12b+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-6\right)^{2}=-24$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-6\right)^{2}}=\sqrt{-24}$$

Simplify.

$$b-6=2\sqrt{6}i b-6=-2\sqrt{6}i$$

Add 6 to both sides of the equation.

$$b=6+2\sqrt{6}i b=-2\sqrt{6}i+6$$