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

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

Solve for b

$$b=10$$

$$b=0$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(b+4\right)^{2}.

$$b^{2}+8b+16=b^{2}+\left(b-4\right)^{2}+b\left(b-4\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(b-4\right)^{2}.

$$b^{2}+8b+16=b^{2}+b^{2}-8b+16+b\left(b-4\right)$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$b^{2}+8b+16=2b^{2}-8b+16+b\left(b-4\right)$$

Use the distributive property to multiply b by b-4.

$$b^{2}+8b+16=2b^{2}-8b+16+b^{2}-4b$$

Combine 2b^{2} and b^{2} to get 3b^{2}.

$$b^{2}+8b+16=3b^{2}-8b+16-4b$$

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

$$b^{2}+8b+16=3b^{2}-12b+16$$

Subtract 3b^{2} from both sides.

$$b^{2}+8b+16-3b^{2}=-12b+16$$

Combine b^{2} and -3b^{2} to get -2b^{2}.

$$-2b^{2}+8b+16=-12b+16$$

Add 12b to both sides.

$$-2b^{2}+8b+16+12b=16$$

Combine 8b and 12b to get 20b.

$$-2b^{2}+20b+16=16$$

Subtract 16 from both sides.

$$-2b^{2}+20b+16-16=0$$

Subtract 16 from 16 to get 0.

$$-2b^{2}+20b=0$$

Factor out b.

$$b\left(-2b+20\right)=0$$

To find equation solutions, solve b=0 and -2b+20=0.

$$b=0 b=10$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(b+4\right)^{2}.

$$b^{2}+8b+16=b^{2}+\left(b-4\right)^{2}+b\left(b-4\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(b-4\right)^{2}.

$$b^{2}+8b+16=b^{2}+b^{2}-8b+16+b\left(b-4\right)$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$b^{2}+8b+16=2b^{2}-8b+16+b\left(b-4\right)$$

Use the distributive property to multiply b by b-4.

$$b^{2}+8b+16=2b^{2}-8b+16+b^{2}-4b$$

Combine 2b^{2} and b^{2} to get 3b^{2}.

$$b^{2}+8b+16=3b^{2}-8b+16-4b$$

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

$$b^{2}+8b+16=3b^{2}-12b+16$$

Subtract 3b^{2} from both sides.

$$b^{2}+8b+16-3b^{2}=-12b+16$$

Combine b^{2} and -3b^{2} to get -2b^{2}.

$$-2b^{2}+8b+16=-12b+16$$

Add 12b to both sides.

$$-2b^{2}+8b+16+12b=16$$

Combine 8b and 12b to get 20b.

$$-2b^{2}+20b+16=16$$

Subtract 16 from both sides.

$$-2b^{2}+20b+16-16=0$$

Subtract 16 from 16 to get 0.

$$-2b^{2}+20b=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 20 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-20±\sqrt{20^{2}}}{2\left(-2\right)}$$

Take the square root of 20^{2}.

$$b=\frac{-20±20}{2\left(-2\right)}$$

Multiply 2 times -2.

$$b=\frac{-20±20}{-4}$$

Now solve the equation b=\frac{-20±20}{-4} when ± is plus. Add -20 to 20.

$$b=\frac{0}{-4}$$

Divide 0 by -4.

$$b=0$$

Now solve the equation b=\frac{-20±20}{-4} when ± is minus. Subtract 20 from -20.

$$b=-\frac{40}{-4}$$

Divide -40 by -4.

$$b=10$$

The equation is now solved.

$$b=0 b=10$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(b+4\right)^{2}.

$$b^{2}+8b+16=b^{2}+\left(b-4\right)^{2}+b\left(b-4\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(b-4\right)^{2}.

$$b^{2}+8b+16=b^{2}+b^{2}-8b+16+b\left(b-4\right)$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$b^{2}+8b+16=2b^{2}-8b+16+b\left(b-4\right)$$

Use the distributive property to multiply b by b-4.

$$b^{2}+8b+16=2b^{2}-8b+16+b^{2}-4b$$

Combine 2b^{2} and b^{2} to get 3b^{2}.

$$b^{2}+8b+16=3b^{2}-8b+16-4b$$

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

$$b^{2}+8b+16=3b^{2}-12b+16$$

Subtract 3b^{2} from both sides.

$$b^{2}+8b+16-3b^{2}=-12b+16$$

Combine b^{2} and -3b^{2} to get -2b^{2}.

$$-2b^{2}+8b+16=-12b+16$$

Add 12b to both sides.

$$-2b^{2}+8b+16+12b=16$$

Combine 8b and 12b to get 20b.

$$-2b^{2}+20b+16=16$$

Subtract 16 from both sides.

$$-2b^{2}+20b=16-16$$

Subtract 16 from 16 to get 0.

$$-2b^{2}+20b=0$$

Divide both sides by -2.

$$\frac{-2b^{2}+20b}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$b^{2}+\frac{20}{-2}b=\frac{0}{-2}$$

Divide 20 by -2.

$$b^{2}-10b=\frac{0}{-2}$$

Divide 0 by -2.

$$b^{2}-10b=0$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-10b+\left(-5\right)^{2}=\left(-5\right)^{2}$$

Square -5.

$$b^{2}-10b+25=25$$

Factor b^{2}-10b+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-5\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-5\right)^{2}}=\sqrt{25}$$

Simplify.

$$b-5=5 b-5=-5$$

Add 5 to both sides of the equation.

$$b=10 b=0$$