Solve 2000(1-x)*(6.5-7.5x)*13%*3=936

$$2000(1-x) \times (6.5-7.5x) \times 13 \% \times 3=936$$

Solve for x

$$x = \frac{\sqrt{37} + 14}{15} \approx 1.338850835$$

$$x=\frac{14-\sqrt{37}}{15}\approx 0.527815831$$

Steps Using the Quadratic Formula

Multiply 2000 and \frac{13}{100} to get 260.

$$260\left(1-x\right)\left(6.5-7.5x\right)\times 3=936$$

Multiply 260 and 3 to get 780.

$$780\left(1-x\right)\left(6.5-7.5x\right)=936$$

Use the distributive property to multiply 780 by 1-x.

$$\left(780-780x\right)\left(6.5-7.5x\right)=936$$

Use the distributive property to multiply 780-780x by 6.5-7.5x and combine like terms.

$$5070-10920x+5850x^{2}=936$$

Subtract 936 from both sides.

$$5070-10920x+5850x^{2}-936=0$$

Subtract 936 from 5070 to get 4134.

$$4134-10920x+5850x^{2}=0$$

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.

$$5850x^{2}-10920x+4134=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5850 for a, -10920 for b, and 4134 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10920\right)±\sqrt{\left(-10920\right)^{2}-4\times 5850\times 4134}}{2\times 5850}$$

Square -10920.

$$x=\frac{-\left(-10920\right)±\sqrt{119246400-4\times 5850\times 4134}}{2\times 5850}$$

Multiply -4 times 5850.

$$x=\frac{-\left(-10920\right)±\sqrt{119246400-23400\times 4134}}{2\times 5850}$$

Multiply -23400 times 4134.

$$x=\frac{-\left(-10920\right)±\sqrt{119246400-96735600}}{2\times 5850}$$

Add 119246400 to -96735600.

$$x=\frac{-\left(-10920\right)±\sqrt{22510800}}{2\times 5850}$$

Take the square root of 22510800.

$$x=\frac{-\left(-10920\right)±780\sqrt{37}}{2\times 5850}$$

The opposite of -10920 is 10920.

$$x=\frac{10920±780\sqrt{37}}{2\times 5850}$$

Multiply 2 times 5850.

$$x=\frac{10920±780\sqrt{37}}{11700}$$

Now solve the equation x=\frac{10920±780\sqrt{37}}{11700} when ± is plus. Add 10920 to 780\sqrt{37}.

$$x=\frac{780\sqrt{37}+10920}{11700}$$

Divide 10920+780\sqrt{37} by 11700.

$$x=\frac{\sqrt{37}+14}{15}$$

Now solve the equation x=\frac{10920±780\sqrt{37}}{11700} when ± is minus. Subtract 780\sqrt{37} from 10920.

$$x=\frac{10920-780\sqrt{37}}{11700}$$

Divide 10920-780\sqrt{37} by 11700.

$$x=\frac{14-\sqrt{37}}{15}$$

The equation is now solved.

$$x=\frac{\sqrt{37}+14}{15} x=\frac{14-\sqrt{37}}{15}$$

Steps for Completing the Square

Multiply 2000 and \frac{13}{100} to get 260.

$$260\left(1-x\right)\left(6.5-7.5x\right)\times 3=936$$

Multiply 260 and 3 to get 780.

$$780\left(1-x\right)\left(6.5-7.5x\right)=936$$

Use the distributive property to multiply 780 by 1-x.

$$\left(780-780x\right)\left(6.5-7.5x\right)=936$$

Use the distributive property to multiply 780-780x by 6.5-7.5x and combine like terms.

$$5070-10920x+5850x^{2}=936$$

Subtract 5070 from both sides.

$$-10920x+5850x^{2}=936-5070$$

Subtract 5070 from 936 to get -4134.

$$-10920x+5850x^{2}=-4134$$

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.

$$5850x^{2}-10920x=-4134$$

Divide both sides by 5850.

$$\frac{5850x^{2}-10920x}{5850}=-\frac{4134}{5850}$$

Dividing by 5850 undoes the multiplication by 5850.

$$x^{2}+\left(-\frac{10920}{5850}\right)x=-\frac{4134}{5850}$$

Reduce the fraction \frac{-10920}{5850} to lowest terms by extracting and canceling out 390.

$$x^{2}-\frac{28}{15}x=-\frac{4134}{5850}$$

Reduce the fraction \frac{-4134}{5850} to lowest terms by extracting and canceling out 78.

$$x^{2}-\frac{28}{15}x=-\frac{53}{75}$$

Divide -\frac{28}{15}, the coefficient of the x term, by 2 to get -\frac{14}{15}. Then add the square of -\frac{14}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{28}{15}x+\left(-\frac{14}{15}\right)^{2}=-\frac{53}{75}+\left(-\frac{14}{15}\right)^{2}$$

Square -\frac{14}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=-\frac{53}{75}+\frac{196}{225}$$

Add -\frac{53}{75} to \frac{196}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=\frac{37}{225}$$

Factor x^{2}-\frac{28}{15}x+\frac{196}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{14}{15}\right)^{2}=\frac{37}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{14}{15}\right)^{2}}=\sqrt{\frac{37}{225}}$$

Simplify.

$$x-\frac{14}{15}=\frac{\sqrt{37}}{15} x-\frac{14}{15}=-\frac{\sqrt{37}}{15}$$

Add \frac{14}{15} to both sides of the equation.

$$x=\frac{\sqrt{37}+14}{15} x=\frac{14-\sqrt{37}}{15}$$