Solve (b+4)=b^2-11

$$( b + 4 ) = b ^ { 2 } - 11$$

Solve for b

$$b = \frac{\sqrt{61} + 1}{2} \approx 4.405124838$$

$$b=\frac{1-\sqrt{61}}{2}\approx -3.405124838$$

Steps Using the Quadratic Formula

Subtract b^{2} from both sides.

$$b+4-b^{2}=-11$$

Add 11 to both sides.

$$b+4-b^{2}+11=0$$

Add 4 and 11 to get 15.

$$b+15-b^{2}=0$$

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.

$$-b^{2}+b+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Square 1.

$$b=\frac{-1±\sqrt{1-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$b=\frac{-1±\sqrt{1+4\times 15}}{2\left(-1\right)}$$

Multiply 4 times 15.

$$b=\frac{-1±\sqrt{1+60}}{2\left(-1\right)}$$

Add 1 to 60.

$$b=\frac{-1±\sqrt{61}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$b=\frac{-1±\sqrt{61}}{-2}$$

Now solve the equation b=\frac{-1±\sqrt{61}}{-2} when ± is plus. Add -1 to \sqrt{61}.

$$b=\frac{\sqrt{61}-1}{-2}$$

Divide -1+\sqrt{61} by -2.

$$b=\frac{1-\sqrt{61}}{2}$$

Now solve the equation b=\frac{-1±\sqrt{61}}{-2} when ± is minus. Subtract \sqrt{61} from -1.

$$b=\frac{-\sqrt{61}-1}{-2}$$

Divide -1-\sqrt{61} by -2.

$$b=\frac{\sqrt{61}+1}{2}$$

The equation is now solved.

$$b=\frac{1-\sqrt{61}}{2} b=\frac{\sqrt{61}+1}{2}$$

Steps for Completing the Square

Subtract b^{2} from both sides.

$$b+4-b^{2}=-11$$

Subtract 4 from both sides.

$$b-b^{2}=-11-4$$

Subtract 4 from -11 to get -15.

$$b-b^{2}=-15$$

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.

$$-b^{2}+b=-15$$

Divide both sides by -1.

$$\frac{-b^{2}+b}{-1}=-\frac{15}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$b^{2}+\frac{1}{-1}b=-\frac{15}{-1}$$

Divide 1 by -1.

$$b^{2}-b=-\frac{15}{-1}$$

Divide -15 by -1.

$$b^{2}-b=15$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-b+\left(-\frac{1}{2}\right)^{2}=15+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-b+\frac{1}{4}=15+\frac{1}{4}$$

Add 15 to \frac{1}{4}.

$$b^{2}-b+\frac{1}{4}=\frac{61}{4}$$

Factor b^{2}-b+\frac{1}{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}.

$$\left(b-\frac{1}{2}\right)^{2}=\frac{61}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{1}{2}\right)^{2}}=\sqrt{\frac{61}{4}}$$

Simplify.

$$b-\frac{1}{2}=\frac{\sqrt{61}}{2} b-\frac{1}{2}=-\frac{\sqrt{61}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$b=\frac{\sqrt{61}+1}{2} b=\frac{1-\sqrt{61}}{2}$$