Solve 84(θ-20)*200/1000*4200*(θ-25)=1.1*420*(100-θ)

$$84( \theta -20) \times \frac{ 200 }{ 1000 } \times 4200 \times ( \theta -25)=1.1 \times 420 \times (100- \theta )$$

Solve for θ

$$\theta =\frac{\sqrt{76288921}+75589}{3360}\approx 25.096235989$$

$$\theta =\frac{75589-\sqrt{76288921}}{3360}\approx 19.897216392$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{200}{1000} to lowest terms by extracting and canceling out 200.

$$84\left(\theta -20\right)\times \frac{1}{5}\times 4200\left(\theta -25\right)=1.1\times 420\left(100-\theta \right)$$

Multiply 84 and \frac{1}{5} to get \frac{84}{5}.

$$\frac{84}{5}\left(\theta -20\right)\times 4200\left(\theta -25\right)=1.1\times 420\left(100-\theta \right)$$

Multiply \frac{84}{5} and 4200 to get 70560.

$$70560\left(\theta -20\right)\left(\theta -25\right)=1.1\times 420\left(100-\theta \right)$$

Use the distributive property to multiply 70560 by \theta -20.

$$\left(70560\theta -1411200\right)\left(\theta -25\right)=1.1\times 420\left(100-\theta \right)$$

Use the distributive property to multiply 70560\theta -1411200 by \theta -25 and combine like terms.

$$70560\theta ^{2}-3175200\theta [phone]=1.1\times 420\left(100-\theta \right)$$

Multiply 1.1 and 420 to get 462.

$$70560\theta ^{2}-3175200\theta [phone]=462\left(100-\theta \right)$$

Use the distributive property to multiply 462 by 100-\theta .

$$70560\theta ^{2}-3175200\theta [phone]=46200-462\theta$$

Subtract 46200 from both sides.

$$70560\theta ^{2}-3175200\theta [phone]=-462\theta$$

Subtract 46200 from 35280000 to get 35233800.

$$70560\theta ^{2}-3175200\theta [phone]=-462\theta$$

Add 462\theta to both sides.

$$70560\theta ^{2}-3175200\theta [phone]+462\theta =0$$

Combine -3175200\theta and 462\theta to get -3174738\theta .

$$70560\theta ^{2}-3174738\theta [phone]=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 70560 for a, -3174738 for b, and 35233800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\theta =\frac{-\left(-3174738\right)±\sqrt{\left(-3174738\right)^{2}-4\times 70560\times 35233800}}{2\times 70560}$$

Square -3174738.

$$\theta =\frac{-\left(-3174738\right)±\sqrt{10078961368644-4\times 70560\times 35233800}}{2\times 70560}$$

Multiply -4 times 70560.

$$\theta =\frac{-\left(-3174738\right)±\sqrt{10078961368644-282240\times 35233800}}{2\times 70560}$$

Multiply -282240 times 35233800.

$$\theta =\frac{-\left(-3174738\right)±\sqrt{10078961368644-9944387712000}}{2\times 70560}$$

Add 10078961368644 to -9944387712000.

$$\theta =\frac{-\left(-3174738\right)±\sqrt{134573656644}}{2\times 70560}$$

Take the square root of 134573656644.

$$\theta =\frac{-\left(-3174738\right)±42\sqrt{76288921}}{2\times 70560}$$

The opposite of -3174738 is 3174738.

$$\theta =\frac{3174738±42\sqrt{76288921}}{2\times 70560}$$

Multiply 2 times 70560.

$$\theta =\frac{3174738±42\sqrt{76288921}}{141120}$$

Now solve the equation \theta =\frac{3174738±42\sqrt{76288921}}{141120} when ± is plus. Add 3174738 to 42\sqrt{76288921}.

$$\theta =\frac{42\sqrt{76288921}+3174738}{141120}$$

Divide 3174738+42\sqrt{76288921} by 141120.

$$\theta =\frac{\sqrt{76288921}+75589}{3360}$$

Now solve the equation \theta =\frac{3174738±42\sqrt{76288921}}{141120} when ± is minus. Subtract 42\sqrt{76288921} from 3174738.

$$\theta =\frac{3174738-42\sqrt{76288921}}{141120}$$

Divide 3174738-42\sqrt{76288921} by 141120.

$$\theta =\frac{75589-\sqrt{76288921}}{3360}$$

The equation is now solved.

$$\theta =\frac{\sqrt{76288921}+75589}{3360} \theta =\frac{75589-\sqrt{76288921}}{3360}$$

Steps for Completing the Square

Reduce the fraction \frac{200}{1000} to lowest terms by extracting and canceling out 200.

$$84\left(\theta -20\right)\times \frac{1}{5}\times 4200\left(\theta -25\right)=1.1\times 420\left(100-\theta \right)$$

Multiply 84 and \frac{1}{5} to get \frac{84}{5}.

$$\frac{84}{5}\left(\theta -20\right)\times 4200\left(\theta -25\right)=1.1\times 420\left(100-\theta \right)$$

Multiply \frac{84}{5} and 4200 to get 70560.

$$70560\left(\theta -20\right)\left(\theta -25\right)=1.1\times 420\left(100-\theta \right)$$

Use the distributive property to multiply 70560 by \theta -20.

$$\left(70560\theta -1411200\right)\left(\theta -25\right)=1.1\times 420\left(100-\theta \right)$$

Use the distributive property to multiply 70560\theta -1411200 by \theta -25 and combine like terms.

$$70560\theta ^{2}-3175200\theta [phone]=1.1\times 420\left(100-\theta \right)$$

Multiply 1.1 and 420 to get 462.

$$70560\theta ^{2}-3175200\theta [phone]=462\left(100-\theta \right)$$

Use the distributive property to multiply 462 by 100-\theta .

$$70560\theta ^{2}-3175200\theta [phone]=46200-462\theta$$

Add 462\theta to both sides.

$$70560\theta ^{2}-3175200\theta [phone]+462\theta =46200$$

Combine -3175200\theta and 462\theta to get -3174738\theta .

$$70560\theta ^{2}-3174738\theta [phone]=46200$$

Subtract 35280000 from both sides.

$$70560\theta ^{2}-3174738\theta =46200-35280000$$

Subtract 35280000 from 46200 to get -35233800.

$$70560\theta ^{2}-3174738\theta =-35233800$$

Divide both sides by 70560.

$$\frac{70560\theta ^{2}-3174738\theta }{70560}=-\frac{35233800}{70560}$$

Dividing by 70560 undoes the multiplication by 70560.

$$\theta ^{2}+\left(-\frac{3174738}{70560}\right)\theta =-\frac{35233800}{70560}$$

Reduce the fraction \frac{-3174738}{70560} to lowest terms by extracting and canceling out 42.

$$\theta ^{2}-\frac{75589}{1680}\theta =-\frac{35233800}{70560}$$

Reduce the fraction \frac{-35233800}{70560} to lowest terms by extracting and canceling out 840.

$$\theta ^{2}-\frac{75589}{1680}\theta =-\frac{41945}{84}$$

Divide -\frac{75589}{1680}, the coefficient of the x term, by 2 to get -\frac{75589}{3360}. Then add the square of -\frac{75589}{3360} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\theta ^{2}-\frac{75589}{1680}\theta +\left(-\frac{75589}{3360}\right)^{2}=-\frac{41945}{84}+\left(-\frac{75589}{3360}\right)^{2}$$

Square -\frac{75589}{3360} by squaring both the numerator and the denominator of the fraction.

$$\theta ^{2}-\frac{75589}{1680}\theta +\frac{5713696921}{11289600}=-\frac{41945}{84}+\frac{5713696921}{11289600}$$

Add -\frac{41945}{84} to \frac{5713696921}{11289600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\theta ^{2}-\frac{75589}{1680}\theta +\frac{5713696921}{11289600}=\frac{76288921}{11289600}$$

Factor \theta ^{2}-\frac{75589}{1680}\theta +\frac{5713696921}{11289600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\theta -\frac{75589}{3360}\right)^{2}=\frac{76288921}{11289600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\theta -\frac{75589}{3360}\right)^{2}}=\sqrt{\frac{76288921}{11289600}}$$

Simplify.

$$\theta -\frac{75589}{3360}=\frac{\sqrt{76288921}}{3360} \theta -\frac{75589}{3360}=-\frac{\sqrt{76288921}}{3360}$$

Add \frac{75589}{3360} to both sides of the equation.

$$\theta =\frac{\sqrt{76288921}+75589}{3360} \theta =\frac{75589-\sqrt{76288921}}{3360}$$