Solve 45=4*(n-1)n/2

$$45 = \frac { 4 \times ( n - 1 ) n } { 2 }$$

Solve for n

$$n = \frac{\sqrt{91} + 1}{2} \approx 5.269696007$$

$$n=\frac{1-\sqrt{91}}{2}\approx -4.269696007$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$45\times 2=4\left(n-1\right)n$$

Multiply 45 and 2 to get 90.

$$90=4\left(n-1\right)n$$

Use the distributive property to multiply 4 by n-1.

$$90=\left(4n-4\right)n$$

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

$$90=4n^{2}-4n$$

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

$$4n^{2}-4n=90$$

Subtract 90 from both sides.

$$4n^{2}-4n-90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -4 for b, and -90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4\left(-90\right)}}{2\times 4}$$

Square -4.

$$n=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-90\right)}}{2\times 4}$$

Multiply -4 times 4.

$$n=\frac{-\left(-4\right)±\sqrt{16-16\left(-90\right)}}{2\times 4}$$

Multiply -16 times -90.

$$n=\frac{-\left(-4\right)±\sqrt{16+1440}}{2\times 4}$$

Add 16 to 1440.

$$n=\frac{-\left(-4\right)±\sqrt{1456}}{2\times 4}$$

Take the square root of 1456.

$$n=\frac{-\left(-4\right)±4\sqrt{91}}{2\times 4}$$

The opposite of -4 is 4.

$$n=\frac{4±4\sqrt{91}}{2\times 4}$$

Multiply 2 times 4.

$$n=\frac{4±4\sqrt{91}}{8}$$

Now solve the equation n=\frac{4±4\sqrt{91}}{8} when ± is plus. Add 4 to 4\sqrt{91}.

$$n=\frac{4\sqrt{91}+4}{8}$$

Divide 4+4\sqrt{91} by 8.

$$n=\frac{\sqrt{91}+1}{2}$$

Now solve the equation n=\frac{4±4\sqrt{91}}{8} when ± is minus. Subtract 4\sqrt{91} from 4.

$$n=\frac{4-4\sqrt{91}}{8}$$

Divide 4-4\sqrt{91} by 8.

$$n=\frac{1-\sqrt{91}}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{91}+1}{2} n=\frac{1-\sqrt{91}}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$45\times 2=4\left(n-1\right)n$$

Multiply 45 and 2 to get 90.

$$90=4\left(n-1\right)n$$

Use the distributive property to multiply 4 by n-1.

$$90=\left(4n-4\right)n$$

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

$$90=4n^{2}-4n$$

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

$$4n^{2}-4n=90$$

Divide both sides by 4.

$$\frac{4n^{2}-4n}{4}=\frac{90}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$n^{2}+\left(-\frac{4}{4}\right)n=\frac{90}{4}$$

Divide -4 by 4.

$$n^{2}-n=\frac{90}{4}$$

Reduce the fraction \frac{90}{4} to lowest terms by extracting and canceling out 2.

$$n^{2}-n=\frac{45}{2}$$

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.

$$n^{2}-n+\left(-\frac{1}{2}\right)^{2}=\frac{45}{2}+\left(-\frac{1}{2}\right)^{2}$$

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

$$n^{2}-n+\frac{1}{4}=\frac{45}{2}+\frac{1}{4}$$

Add \frac{45}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-n+\frac{1}{4}=\frac{91}{4}$$

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

$$\left(n-\frac{1}{2}\right)^{2}=\frac{91}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{1}{2}\right)^{2}}=\sqrt{\frac{91}{4}}$$

Simplify.

$$n-\frac{1}{2}=\frac{\sqrt{91}}{2} n-\frac{1}{2}=-\frac{\sqrt{91}}{2}$$

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

$$n=\frac{\sqrt{91}+1}{2} n=\frac{1-\sqrt{91}}{2}$$