Solve 3-20001/2350*3.8310^-7100left(y-0.7right)^2=2.5

$$3-2000 \frac{ 1 }{ 2 } 350 \cdot 3.83 { 10 }^{ -7 } 100 { \left(y-0.7 \right) }^{ 2 } = 2.5$$

Solve for y

$$y=\frac{200\sqrt{53633405}}{10726681}+0.7\approx 0.836547048$$

$$y=-\frac{200\sqrt{53633405}}{10726681}+0.7\approx 0.563452952$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$6-\left(2000\times 2+1\right)\times 350\times 3.83\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Multiply 2000 and 2 to get 4000.

$$6-\left(4000+1\right)\times 350\times 3.83\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Add 4000 and 1 to get 4001.

$$6-4001\times 350\times 3.83\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Multiply 4001 and 350 to get 1400350.

$$6-1400350\times 3.83\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Multiply 1400350 and 3.83 to get 5363340.5.

$$6-5363340.5\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$6-5363340.5\times \frac{1}{10000000}\times 100\left(y-0.7\right)^{2}=5$$

Multiply 5363340.5 and \frac{1}{10000000} to get \frac{10726681}{20000000}.

$$6-\frac{10726681}{20000000}\times 100\left(y-0.7\right)^{2}=5$$

Multiply \frac{10726681}{20000000} and 100 to get \frac{10726681}{200000}.

$$6-\frac{10726681}{200000}\left(y-0.7\right)^{2}=5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-0.7\right)^{2}.

$$6-\frac{10726681}{200000}\left(y^{2}-1.4y+0.49\right)=5$$

Use the distributive property to multiply -\frac{10726681}{200000} by y^{2}-1.4y+0.49.

$$6-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y-\frac{525607369}{20000000}=5$$

Subtract \frac{525607369}{20000000} from 6 to get -\frac{405607369}{20000000}.

$$-\frac{405607369}{20000000}-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y=5$$

Subtract 5 from both sides.

$$-\frac{405607369}{20000000}-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y-5=0$$

Subtract 5 from -\frac{405607369}{20000000} to get -\frac{505607369}{20000000}.

$$-\frac{505607369}{20000000}-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y=0$$

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.

$$-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y-\frac{505607369}{20000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{10726681}{200000} for a, \frac{75086767}{1000000} for b, and -\frac{505607369}{20000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{75086767}{1000000}±\sqrt{\left(\frac{75086767}{1000000}\right)^{2}-4\left(-\frac{10726681}{200000}\right)\left(-\frac{505607369}{20000000}\right)}}{2\left(-\frac{10726681}{200000}\right)}$$

Square \frac{75086767}{1000000} by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\frac{75086767}{1000000}±\sqrt{\frac{5638022578512289}{1000000000000}-4\left(-\frac{10726681}{200000}\right)\left(-\frac{505607369}{20000000}\right)}}{2\left(-\frac{10726681}{200000}\right)}$$

Multiply -4 times -\frac{10726681}{200000}.

$$y=\frac{-\frac{75086767}{1000000}±\sqrt{\frac{5638022578512289}{1000000000000}+\frac{10726681}{50000}\left(-\frac{505607369}{20000000}\right)}}{2\left(-\frac{10726681}{200000}\right)}$$

Multiply \frac{10726681}{50000} times -\frac{505607369}{20000000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{75086767}{1000000}±\sqrt{\frac{5638022578512289-5423488958512289}{1000000000000}}}{2\left(-\frac{10726681}{200000}\right)}$$

Add \frac{5638022578512289}{1000000000000} to -\frac{5423488958512289}{1000000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{75086767}{1000000}±\sqrt{\frac{10726681}{50000}}}{2\left(-\frac{10726681}{200000}\right)}$$

Take the square root of \frac{10726681}{50000}.

$$y=\frac{-\frac{75086767}{1000000}±\frac{\sqrt{53633405}}{500}}{2\left(-\frac{10726681}{200000}\right)}$$

Multiply 2 times -\frac{10726681}{200000}.

$$y=\frac{-\frac{75086767}{1000000}±\frac{\sqrt{53633405}}{500}}{-\frac{10726681}{100000}}$$

Now solve the equation y=\frac{-\frac{75086767}{1000000}±\frac{\sqrt{53633405}}{500}}{-\frac{10726681}{100000}} when ± is plus. Add -\frac{75086767}{1000000} to \frac{\sqrt{53633405}}{500}.

$$y=\frac{\frac{\sqrt{53633405}}{500}-\frac{75086767}{1000000}}{-\frac{10726681}{100000}}$$

Divide -\frac{75086767}{1000000}+\frac{\sqrt{53633405}}{500} by -\frac{10726681}{100000} by multiplying -\frac{75086767}{1000000}+\frac{\sqrt{53633405}}{500} by the reciprocal of -\frac{10726681}{100000}.

$$y=-\frac{200\sqrt{53633405}}{10726681}+\frac{7}{10}$$

Now solve the equation y=\frac{-\frac{75086767}{1000000}±\frac{\sqrt{53633405}}{500}}{-\frac{10726681}{100000}} when ± is minus. Subtract \frac{\sqrt{53633405}}{500} from -\frac{75086767}{1000000}.

$$y=\frac{-\frac{\sqrt{53633405}}{500}-\frac{75086767}{1000000}}{-\frac{10726681}{100000}}$$

Divide -\frac{75086767}{1000000}-\frac{\sqrt{53633405}}{500} by -\frac{10726681}{100000} by multiplying -\frac{75086767}{1000000}-\frac{\sqrt{53633405}}{500} by the reciprocal of -\frac{10726681}{100000}.

$$y=\frac{200\sqrt{53633405}}{10726681}+\frac{7}{10}$$

The equation is now solved.

$$y=-\frac{200\sqrt{53633405}}{10726681}+\frac{7}{10} y=\frac{200\sqrt{53633405}}{10726681}+\frac{7}{10}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$6-\left(2000\times 2+1\right)\times 350\times 3.83\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Multiply 2000 and 2 to get 4000.

$$6-\left(4000+1\right)\times 350\times 3.83\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Add 4000 and 1 to get 4001.

$$6-4001\times 350\times 3.83\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Multiply 4001 and 350 to get 1400350.

$$6-1400350\times 3.83\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Multiply 1400350 and 3.83 to get 5363340.5.

$$6-5363340.5\times 10^{-7}\times 100\left(y-0.7\right)^{2}=5$$

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$6-5363340.5\times \frac{1}{10000000}\times 100\left(y-0.7\right)^{2}=5$$

Multiply 5363340.5 and \frac{1}{10000000} to get \frac{10726681}{20000000}.

$$6-\frac{10726681}{20000000}\times 100\left(y-0.7\right)^{2}=5$$

Multiply \frac{10726681}{20000000} and 100 to get \frac{10726681}{200000}.

$$6-\frac{10726681}{200000}\left(y-0.7\right)^{2}=5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-0.7\right)^{2}.

$$6-\frac{10726681}{200000}\left(y^{2}-1.4y+0.49\right)=5$$

Use the distributive property to multiply -\frac{10726681}{200000} by y^{2}-1.4y+0.49.

$$6-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y-\frac{525607369}{20000000}=5$$

Subtract \frac{525607369}{20000000} from 6 to get -\frac{405607369}{20000000}.

$$-\frac{405607369}{20000000}-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y=5$$

Add \frac{405607369}{20000000} to both sides.

$$-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y=5+\frac{405607369}{20000000}$$

Add 5 and \frac{405607369}{20000000} to get \frac{505607369}{20000000}.

$$-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y=\frac{505607369}{20000000}$$

Divide both sides of the equation by -\frac{10726681}{200000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{10726681}{200000}y^{2}+\frac{75086767}{1000000}y}{-\frac{10726681}{200000}}=\frac{\frac{505607369}{20000000}}{-\frac{10726681}{200000}}$$

Dividing by -\frac{10726681}{200000} undoes the multiplication by -\frac{10726681}{200000}.

$$y^{2}+\frac{\frac{75086767}{1000000}}{-\frac{10726681}{200000}}y=\frac{\frac{505607369}{20000000}}{-\frac{10726681}{200000}}$$

Divide \frac{75086767}{1000000} by -\frac{10726681}{200000} by multiplying \frac{75086767}{1000000} by the reciprocal of -\frac{10726681}{200000}.

$$y^{2}-\frac{7}{5}y=\frac{\frac{505607369}{20000000}}{-\frac{10726681}{200000}}$$

Divide \frac{505607369}{20000000} by -\frac{10726681}{200000} by multiplying \frac{505607369}{20000000} by the reciprocal of -\frac{10726681}{200000}.

$$y^{2}-\frac{7}{5}y=-\frac{505607369}{1072668100}$$

Divide -\frac{7}{5}, the coefficient of the x term, by 2 to get -\frac{7}{10}. Then add the square of -\frac{7}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{7}{5}y+\left(-\frac{7}{10}\right)^{2}=-\frac{505607369}{1072668100}+\left(-\frac{7}{10}\right)^{2}$$

Square -\frac{7}{10} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{7}{5}y+\frac{49}{100}=-\frac{505607369}{1072668100}+\frac{49}{100}$$

Add -\frac{505607369}{1072668100} to \frac{49}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{7}{5}y+\frac{49}{100}=\frac{200000}{10726681}$$

Factor y^{2}-\frac{7}{5}y+\frac{49}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{7}{10}\right)^{2}=\frac{200000}{10726681}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{7}{10}\right)^{2}}=\sqrt{\frac{200000}{10726681}}$$

Simplify.

$$y-\frac{7}{10}=\frac{200\sqrt{53633405}}{10726681} y-\frac{7}{10}=-\frac{200\sqrt{53633405}}{10726681}$$

Add \frac{7}{10} to both sides of the equation.

$$y=\frac{200\sqrt{53633405}}{10726681}+\frac{7}{10} y=-\frac{200\sqrt{53633405}}{10726681}+\frac{7}{10}$$