Solve 2500=75L_{2}+L_2^2

$$2500 = 75 L _ { 2 } + L _ { 2 } ^ { 2 }$$

Solve for L_2

$$L_{2}=-100$$

$$L_{2}=25$$

Steps Using Factoring

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

$$75L_{2}+L_{2}^{2}=2500$$

Subtract 2500 from both sides.

$$75L_{2}+L_{2}^{2}-2500=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$L_{2}^{2}+75L_{2}-2500=0$$

To solve the equation, factor L_{2}^{2}+75L_{2}-2500 using formula L_{2}^{2}+\left(a+b\right)L_{2}+ab=\left(L_{2}+a\right)\left(L_{2}+b\right). To find a and b, set up a system to be solved.

$$a+b=75 ab=-2500$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2500.

$$-1,2500 -2,1250 -4,625 -5,500 -10,250 -20,125 -25,100 -50,50$$

Calculate the sum for each pair.

$$-1+2500=2499 -2+1250=1248 -4+625=621 -5+500=495 -10+250=240 -20+125=105 -25+100=75 -50+50=0$$

The solution is the pair that gives sum 75.

$$a=-25 b=100$$

Rewrite factored expression \left(L_{2}+a\right)\left(L_{2}+b\right) using the obtained values.

$$\left(L_{2}-25\right)\left(L_{2}+100\right)$$

To find equation solutions, solve L_{2}-25=0 and L_{2}+100=0.

$$L_{2}=25 L_{2}=-100$$

Steps Using Factoring By Grouping

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

$$75L_{2}+L_{2}^{2}=2500$$

Subtract 2500 from both sides.

$$75L_{2}+L_{2}^{2}-2500=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$L_{2}^{2}+75L_{2}-2500=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as L_{2}^{2}+aL_{2}+bL_{2}-2500. To find a and b, set up a system to be solved.

$$a+b=75 ab=1\left(-2500\right)=-2500$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2500.

$$-1,2500 -2,1250 -4,625 -5,500 -10,250 -20,125 -25,100 -50,50$$

Calculate the sum for each pair.

$$-1+2500=2499 -2+1250=1248 -4+625=621 -5+500=495 -10+250=240 -20+125=105 -25+100=75 -50+50=0$$

The solution is the pair that gives sum 75.

$$a=-25 b=100$$

Rewrite L_{2}^{2}+75L_{2}-2500 as \left(L_{2}^{2}-25L_{2}\right)+\left(100L_{2}-2500\right).

$$\left(L_{2}^{2}-25L_{2}\right)+\left(100L_{2}-2500\right)$$

Factor out L_{2} in the first and 100 in the second group.

$$L_{2}\left(L_{2}-25\right)+100\left(L_{2}-25\right)$$

Factor out common term L_{2}-25 by using distributive property.

$$\left(L_{2}-25\right)\left(L_{2}+100\right)$$

To find equation solutions, solve L_{2}-25=0 and L_{2}+100=0.

$$L_{2}=25 L_{2}=-100$$

Steps Using the Quadratic Formula

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

$$75L_{2}+L_{2}^{2}=2500$$

Subtract 2500 from both sides.

$$75L_{2}+L_{2}^{2}-2500=0$$

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.

$$L_{2}^{2}+75L_{2}-2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 75 for b, and -2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$L_{2}=\frac{-75±\sqrt{75^{2}-4\left(-2500\right)}}{2}$$

Square 75.

$$L_{2}=\frac{-75±\sqrt{5625-4\left(-2500\right)}}{2}$$

Multiply -4 times -2500.

$$L_{2}=\frac{-75±\sqrt{5625+10000}}{2}$$

Add 5625 to 10000.

$$L_{2}=\frac{-75±\sqrt{15625}}{2}$$

Take the square root of 15625.

$$L_{2}=\frac{-75±125}{2}$$

Now solve the equation L_{2}=\frac{-75±125}{2} when ± is plus. Add -75 to 125.

$$L_{2}=\frac{50}{2}$$

Divide 50 by 2.

$$L_{2}=25$$

Now solve the equation L_{2}=\frac{-75±125}{2} when ± is minus. Subtract 125 from -75.

$$L_{2}=-\frac{200}{2}$$

Divide -200 by 2.

$$L_{2}=-100$$

The equation is now solved.

$$L_{2}=25 L_{2}=-100$$

Steps for Completing the Square

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

$$75L_{2}+L_{2}^{2}=2500$$

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.

$$L_{2}^{2}+75L_{2}=2500$$

Divide 75, the coefficient of the x term, by 2 to get \frac{75}{2}. Then add the square of \frac{75}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$L_{2}^{2}+75L_{2}+\left(\frac{75}{2}\right)^{2}=2500+\left(\frac{75}{2}\right)^{2}$$

Square \frac{75}{2} by squaring both the numerator and the denominator of the fraction.

$$L_{2}^{2}+75L_{2}+\frac{5625}{4}=2500+\frac{5625}{4}$$

Add 2500 to \frac{5625}{4}.

$$L_{2}^{2}+75L_{2}+\frac{5625}{4}=\frac{15625}{4}$$

Factor L_{2}^{2}+75L_{2}+\frac{5625}{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}.

$$\left(L_{2}+\frac{75}{2}\right)^{2}=\frac{15625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(L_{2}+\frac{75}{2}\right)^{2}}=\sqrt{\frac{15625}{4}}$$

Simplify.

$$L_{2}+\frac{75}{2}=\frac{125}{2} L_{2}+\frac{75}{2}=-\frac{125}{2}$$

Subtract \frac{75}{2} from both sides of the equation.

$$L_{2}=25 L_{2}=-100$$