Solve 400=1/2b(b+7)

$$400 = 1 / 2 b ( b + 7 )$$

Solve for b

$$b=-32$$

$$b=25$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2}b by b+7.

$$400=\frac{1}{2}bb+\frac{1}{2}b\times 7$$

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

$$400=\frac{1}{2}b^{2}+\frac{1}{2}b\times 7$$

Multiply \frac{1}{2} and 7 to get \frac{7}{2}.

$$400=\frac{1}{2}b^{2}+\frac{7}{2}b$$

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

$$\frac{1}{2}b^{2}+\frac{7}{2}b=400$$

Subtract 400 from both sides.

$$\frac{1}{2}b^{2}+\frac{7}{2}b-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, \frac{7}{2} for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\frac{7}{2}±\sqrt{\left(\frac{7}{2}\right)^{2}-4\times \frac{1}{2}\left(-400\right)}}{2\times \frac{1}{2}}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$b=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}-4\times \frac{1}{2}\left(-400\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$b=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}-2\left(-400\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -400.

$$b=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}+800}}{2\times \frac{1}{2}}$$

Add \frac{49}{4} to 800.

$$b=\frac{-\frac{7}{2}±\sqrt{\frac{3249}{4}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{3249}{4}.

$$b=\frac{-\frac{7}{2}±\frac{57}{2}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$b=\frac{-\frac{7}{2}±\frac{57}{2}}{1}$$

Now solve the equation b=\frac{-\frac{7}{2}±\frac{57}{2}}{1} when ± is plus. Add -\frac{7}{2} to \frac{57}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$b=\frac{25}{1}$$

Divide 25 by 1.

$$b=25$$

Now solve the equation b=\frac{-\frac{7}{2}±\frac{57}{2}}{1} when ± is minus. Subtract \frac{57}{2} from -\frac{7}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$b=-\frac{32}{1}$$

Divide -32 by 1.

$$b=-32$$

The equation is now solved.

$$b=25 b=-32$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2}b by b+7.

$$400=\frac{1}{2}bb+\frac{1}{2}b\times 7$$

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

$$400=\frac{1}{2}b^{2}+\frac{1}{2}b\times 7$$

Multiply \frac{1}{2} and 7 to get \frac{7}{2}.

$$400=\frac{1}{2}b^{2}+\frac{7}{2}b$$

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

$$\frac{1}{2}b^{2}+\frac{7}{2}b=400$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}b^{2}+\frac{7}{2}b}{\frac{1}{2}}=\frac{400}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$b^{2}+\frac{\frac{7}{2}}{\frac{1}{2}}b=\frac{400}{\frac{1}{2}}$$

Divide \frac{7}{2} by \frac{1}{2} by multiplying \frac{7}{2} by the reciprocal of \frac{1}{2}.

$$b^{2}+7b=\frac{400}{\frac{1}{2}}$$

Divide 400 by \frac{1}{2} by multiplying 400 by the reciprocal of \frac{1}{2}.

$$b^{2}+7b=800$$

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

$$b^{2}+7b+\left(\frac{7}{2}\right)^{2}=800+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$b^{2}+7b+\frac{49}{4}=800+\frac{49}{4}$$

Add 800 to \frac{49}{4}.

$$b^{2}+7b+\frac{49}{4}=\frac{3249}{4}$$

Factor b^{2}+7b+\frac{49}{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}.

$$\left(b+\frac{7}{2}\right)^{2}=\frac{3249}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b+\frac{7}{2}\right)^{2}}=\sqrt{\frac{3249}{4}}$$

Simplify.

$$b+\frac{7}{2}=\frac{57}{2} b+\frac{7}{2}=-\frac{57}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$b=25 b=-32$$