Solve 190000(1+x)^2=110000(1+x)+110000

$$190000 ( 1 + x ) ^ { 2 } = 110000 ( 1 + x ) + 110000$$

Solve for x

$$x=\frac{\sqrt{957}-27}{38}\approx 0.103563595$$

$$x=\frac{-\sqrt{957}-27}{38}\approx -1.524616226$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$190000\left(1+2x+x^{2}\right)=110000\left(1+x\right)+110000$$

Use the distributive property to multiply 190000 by 1+2x+x^{2}.

$$190000+380000x+190000x^{2}=110000\left(1+x\right)+110000$$

Use the distributive property to multiply 110000 by 1+x.

$$190000+380000x+190000x^{2}=110000+110000x+110000$$

Add 110000 and 110000 to get 220000.

$$190000+380000x+190000x^{2}=220000+110000x$$

Subtract 220000 from both sides.

$$190000+380000x+190000x^{2}-220000=110000x$$

Subtract 220000 from 190000 to get -30000.

$$-30000+380000x+190000x^{2}=110000x$$

Subtract 110000x from both sides.

$$-30000+380000x+190000x^{2}-110000x=0$$

Combine 380000x and -110000x to get 270000x.

$$-30000+270000x+190000x^{2}=0$$

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.

$$190000x^{2}+270000x-30000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 190000 for a, 270000 for b, and -30000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-270000±\sqrt{270000^{2}-4\times 190000\left(-30000\right)}}{2\times 190000}$$

Square 270000.

$$x=\frac{-270000±\sqrt{72900000000-4\times 190000\left(-30000\right)}}{2\times 190000}$$

Multiply -4 times 190000.

$$x=\frac{-270000±\sqrt{72900000000-760000\left(-30000\right)}}{2\times 190000}$$

Multiply -760000 times -30000.

$$x=\frac{-270000±\sqrt{72900000000+22800000000}}{2\times 190000}$$

Add 72900000000 to 22800000000.

$$x=\frac{-270000±\sqrt{95700000000}}{2\times 190000}$$

Take the square root of 95700000000.

$$x=\frac{-270000±10000\sqrt{957}}{2\times 190000}$$

Multiply 2 times 190000.

$$x=\frac{-270000±10000\sqrt{957}}{380000}$$

Now solve the equation x=\frac{-270000±10000\sqrt{957}}{380000} when ± is plus. Add -270000 to 10000\sqrt{957}.

$$x=\frac{10000\sqrt{957}-270000}{380000}$$

Divide -270000+10000\sqrt{957} by 380000.

$$x=\frac{\sqrt{957}-27}{38}$$

Now solve the equation x=\frac{-270000±10000\sqrt{957}}{380000} when ± is minus. Subtract 10000\sqrt{957} from -270000.

$$x=\frac{-10000\sqrt{957}-270000}{380000}$$

Divide -270000-10000\sqrt{957} by 380000.

$$x=\frac{-\sqrt{957}-27}{38}$$

The equation is now solved.

$$x=\frac{\sqrt{957}-27}{38} x=\frac{-\sqrt{957}-27}{38}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$190000\left(1+2x+x^{2}\right)=110000\left(1+x\right)+110000$$

Use the distributive property to multiply 190000 by 1+2x+x^{2}.

$$190000+380000x+190000x^{2}=110000\left(1+x\right)+110000$$

Use the distributive property to multiply 110000 by 1+x.

$$190000+380000x+190000x^{2}=110000+110000x+110000$$

Add 110000 and 110000 to get 220000.

$$190000+380000x+190000x^{2}=220000+110000x$$

Subtract 110000x from both sides.

$$190000+380000x+190000x^{2}-110000x=220000$$

Combine 380000x and -110000x to get 270000x.

$$190000+270000x+190000x^{2}=220000$$

Subtract 190000 from both sides.

$$270000x+190000x^{2}=220000-190000$$

Subtract 190000 from 220000 to get 30000.

$$270000x+190000x^{2}=30000$$

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.

$$190000x^{2}+270000x=30000$$

Divide both sides by 190000.

$$\frac{190000x^{2}+270000x}{190000}=\frac{30000}{190000}$$

Dividing by 190000 undoes the multiplication by 190000.

$$x^{2}+\frac{270000}{190000}x=\frac{30000}{190000}$$

Reduce the fraction \frac{270000}{190000} to lowest terms by extracting and canceling out 10000.

$$x^{2}+\frac{27}{19}x=\frac{30000}{190000}$$

Reduce the fraction \frac{30000}{190000} to lowest terms by extracting and canceling out 10000.

$$x^{2}+\frac{27}{19}x=\frac{3}{19}$$

Divide \frac{27}{19}, the coefficient of the x term, by 2 to get \frac{27}{38}. Then add the square of \frac{27}{38} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{27}{19}x+\left(\frac{27}{38}\right)^{2}=\frac{3}{19}+\left(\frac{27}{38}\right)^{2}$$

Square \frac{27}{38} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{27}{19}x+\frac{729}{1444}=\frac{3}{19}+\frac{729}{1444}$$

Add \frac{3}{19} to \frac{729}{1444} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{27}{19}x+\frac{729}{1444}=\frac{957}{1444}$$

Factor x^{2}+\frac{27}{19}x+\frac{729}{1444}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{27}{38}\right)^{2}=\frac{957}{1444}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{27}{38}\right)^{2}}=\sqrt{\frac{957}{1444}}$$

Simplify.

$$x+\frac{27}{38}=\frac{\sqrt{957}}{38} x+\frac{27}{38}=-\frac{\sqrt{957}}{38}$$

Subtract \frac{27}{38} from both sides of the equation.

$$x=\frac{\sqrt{957}-27}{38} x=\frac{-\sqrt{957}-27}{38}$$