Solve [(2*40)+(n-1)10]n/2=2710

$$\frac { [ ( 2 \times 40 ) + ( n - 1 ) 10 ] n } { 2 } = 2710$$

Solve for n

$$n = \frac{\sqrt{2217} - 7}{2} \approx 20.042514734$$

$$n=\frac{-\sqrt{2217}-7}{2}\approx -27.042514734$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$\left(2\times 40+\left(n-1\right)\times 10\right)n=2710\times 2$$

Multiply 2 and 40 to get 80.

$$\left(80+\left(n-1\right)\times 10\right)n=2710\times 2$$

Use the distributive property to multiply n-1 by 10.

$$\left(80+10n-10\right)n=2710\times 2$$

Subtract 10 from 80 to get 70.

$$\left(70+10n\right)n=2710\times 2$$

Use the distributive property to multiply 70+10n by n.

$$70n+10n^{2}=2710\times 2$$

Multiply 2710 and 2 to get 5420.

$$70n+10n^{2}=5420$$

Subtract 5420 from both sides.

$$70n+10n^{2}-5420=0$$

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.

$$10n^{2}+70n-5420=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 70 for b, and -5420 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-70±\sqrt{70^{2}-4\times 10\left(-5420\right)}}{2\times 10}$$

Square 70.

$$n=\frac{-70±\sqrt{4900-4\times 10\left(-5420\right)}}{2\times 10}$$

Multiply -4 times 10.

$$n=\frac{-70±\sqrt{4900-40\left(-5420\right)}}{2\times 10}$$

Multiply -40 times -5420.

$$n=\frac{-70±\sqrt{4900+216800}}{2\times 10}$$

Add 4900 to 216800.

$$n=\frac{-70±\sqrt{221700}}{2\times 10}$$

Take the square root of 221700.

$$n=\frac{-70±10\sqrt{2217}}{2\times 10}$$

Multiply 2 times 10.

$$n=\frac{-70±10\sqrt{2217}}{20}$$

Now solve the equation n=\frac{-70±10\sqrt{2217}}{20} when ± is plus. Add -70 to 10\sqrt{2217}.

$$n=\frac{10\sqrt{2217}-70}{20}$$

Divide -70+10\sqrt{2217} by 20.

$$n=\frac{\sqrt{2217}-7}{2}$$

Now solve the equation n=\frac{-70±10\sqrt{2217}}{20} when ± is minus. Subtract 10\sqrt{2217} from -70.

$$n=\frac{-10\sqrt{2217}-70}{20}$$

Divide -70-10\sqrt{2217} by 20.

$$n=\frac{-\sqrt{2217}-7}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{2217}-7}{2} n=\frac{-\sqrt{2217}-7}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$\left(2\times 40+\left(n-1\right)\times 10\right)n=2710\times 2$$

Multiply 2 and 40 to get 80.

$$\left(80+\left(n-1\right)\times 10\right)n=2710\times 2$$

Use the distributive property to multiply n-1 by 10.

$$\left(80+10n-10\right)n=2710\times 2$$

Subtract 10 from 80 to get 70.

$$\left(70+10n\right)n=2710\times 2$$

Use the distributive property to multiply 70+10n by n.

$$70n+10n^{2}=2710\times 2$$

Multiply 2710 and 2 to get 5420.

$$70n+10n^{2}=5420$$

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.

$$10n^{2}+70n=5420$$

Divide both sides by 10.

$$\frac{10n^{2}+70n}{10}=\frac{5420}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$n^{2}+\frac{70}{10}n=\frac{5420}{10}$$

Divide 70 by 10.

$$n^{2}+7n=\frac{5420}{10}$$

Divide 5420 by 10.

$$n^{2}+7n=542$$

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.

$$n^{2}+7n+\left(\frac{7}{2}\right)^{2}=542+\left(\frac{7}{2}\right)^{2}$$

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

$$n^{2}+7n+\frac{49}{4}=542+\frac{49}{4}$$

Add 542 to \frac{49}{4}.

$$n^{2}+7n+\frac{49}{4}=\frac{2217}{4}$$

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}.

$$\left(n+\frac{7}{2}\right)^{2}=\frac{2217}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{7}{2}\right)^{2}}=\sqrt{\frac{2217}{4}}$$

Simplify.

$$n+\frac{7}{2}=\frac{\sqrt{2217}}{2} n+\frac{7}{2}=-\frac{\sqrt{2217}}{2}$$

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

$$n=\frac{\sqrt{2217}-7}{2} n=\frac{-\sqrt{2217}-7}{2}$$