Solve n(4-n)+n(n-1)/2=0

$$n ( 4 - n ) + \frac { n ( n - 1 ) } { 2 } = 0$$

Solve for n

$$n=7$$

$$n=0$$

Steps Using Factoring

Multiply both sides of the equation by 2.

$$2n\left(4-n\right)+n\left(n-1\right)=0$$

Use the distributive property to multiply 2n by 4-n.

$$8n-2n^{2}+n\left(n-1\right)=0$$

Use the distributive property to multiply n by n-1.

$$8n-2n^{2}+n^{2}-n=0$$

Combine -2n^{2} and n^{2} to get -n^{2}.

$$8n-n^{2}-n=0$$

Combine 8n and -n to get 7n.

$$7n-n^{2}=0$$

Factor out n.

$$n\left(7-n\right)=0$$

To find equation solutions, solve n=0 and 7-n=0.

$$n=0 n=7$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2n\left(4-n\right)+n\left(n-1\right)=0$$

Use the distributive property to multiply 2n by 4-n.

$$8n-2n^{2}+n\left(n-1\right)=0$$

Use the distributive property to multiply n by n-1.

$$8n-2n^{2}+n^{2}-n=0$$

Combine -2n^{2} and n^{2} to get -n^{2}.

$$8n-n^{2}-n=0$$

Combine 8n and -n to get 7n.

$$7n-n^{2}=0$$

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.

$$-n^{2}+7n=0$$

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

$$n=\frac{-7±\sqrt{7^{2}}}{2\left(-1\right)}$$

Take the square root of 7^{2}.

$$n=\frac{-7±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-7±7}{-2}$$

Now solve the equation n=\frac{-7±7}{-2} when ± is plus. Add -7 to 7.

$$n=\frac{0}{-2}$$

Divide 0 by -2.

$$n=0$$

Now solve the equation n=\frac{-7±7}{-2} when ± is minus. Subtract 7 from -7.

$$n=-\frac{14}{-2}$$

Divide -14 by -2.

$$n=7$$

The equation is now solved.

$$n=0 n=7$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2n\left(4-n\right)+n\left(n-1\right)=0$$

Use the distributive property to multiply 2n by 4-n.

$$8n-2n^{2}+n\left(n-1\right)=0$$

Use the distributive property to multiply n by n-1.

$$8n-2n^{2}+n^{2}-n=0$$

Combine -2n^{2} and n^{2} to get -n^{2}.

$$8n-n^{2}-n=0$$

Combine 8n and -n to get 7n.

$$7n-n^{2}=0$$

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.

$$-n^{2}+7n=0$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 7 by -1.

$$n^{2}-7n=\frac{0}{-1}$$

Divide 0 by -1.

$$n^{2}-7n=0$$

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.

$$n^{2}-7n+\left(-\frac{7}{2}\right)^{2}=\left(-\frac{7}{2}\right)^{2}$$

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

$$n^{2}-7n+\frac{49}{4}=\frac{49}{4}$$

Factor n^{2}-7n+\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}.

$$\left(n-\frac{7}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{7}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$n-\frac{7}{2}=\frac{7}{2} n-\frac{7}{2}=-\frac{7}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$n=7 n=0$$