Solve 35x(765-85x)=405

$$35x(765-85x)=405$$

Solve for x

$$x=\frac{9\sqrt{351645}}{1190}+\frac{9}{2}\approx 8.984848442$$

$$x=-\frac{9\sqrt{351645}}{1190}+\frac{9}{2}\approx 0.015151558$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 35x by 765-85x.

$$26775x-2975x^{2}=405$$

Subtract 405 from both sides.

$$26775x-2975x^{2}-405=0$$

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.

$$-2975x^{2}+26775x-405=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2975 for a, 26775 for b, and -405 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-26775±\sqrt{26775^{2}-4\left(-2975\right)\left(-405\right)}}{2\left(-2975\right)}$$

Square 26775.

$$x=\frac{-26775±\sqrt{716900625-4\left(-2975\right)\left(-405\right)}}{2\left(-2975\right)}$$

Multiply -4 times -2975.

$$x=\frac{-26775±\sqrt{716900625+11900\left(-405\right)}}{2\left(-2975\right)}$$

Multiply 11900 times -405.

$$x=\frac{-26775±\sqrt{716900625-4819500}}{2\left(-2975\right)}$$

Add 716900625 to -4819500.

$$x=\frac{-26775±\sqrt{712081125}}{2\left(-2975\right)}$$

Take the square root of 712081125.

$$x=\frac{-26775±45\sqrt{351645}}{2\left(-2975\right)}$$

Multiply 2 times -2975.

$$x=\frac{-26775±45\sqrt{351645}}{-5950}$$

Now solve the equation x=\frac{-26775±45\sqrt{351645}}{-5950} when ± is plus. Add -26775 to 45\sqrt{351645}.

$$x=\frac{45\sqrt{351645}-26775}{-5950}$$

Divide -26775+45\sqrt{351645} by -5950.

$$x=-\frac{9\sqrt{351645}}{1190}+\frac{9}{2}$$

Now solve the equation x=\frac{-26775±45\sqrt{351645}}{-5950} when ± is minus. Subtract 45\sqrt{351645} from -26775.

$$x=\frac{-45\sqrt{351645}-26775}{-5950}$$

Divide -26775-45\sqrt{351645} by -5950.

$$x=\frac{9\sqrt{351645}}{1190}+\frac{9}{2}$$

The equation is now solved.

$$x=-\frac{9\sqrt{351645}}{1190}+\frac{9}{2} x=\frac{9\sqrt{351645}}{1190}+\frac{9}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 35x by 765-85x.

$$26775x-2975x^{2}=405$$

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.

$$-2975x^{2}+26775x=405$$

Divide both sides by -2975.

$$\frac{-2975x^{2}+26775x}{-2975}=\frac{405}{-2975}$$

Dividing by -2975 undoes the multiplication by -2975.

$$x^{2}+\frac{26775}{-2975}x=\frac{405}{-2975}$$

Divide 26775 by -2975.

$$x^{2}-9x=\frac{405}{-2975}$$

Reduce the fraction \frac{405}{-2975} to lowest terms by extracting and canceling out 5.

$$x^{2}-9x=-\frac{81}{595}$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-\frac{81}{595}+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-\frac{81}{595}+\frac{81}{4}$$

Add -\frac{81}{595} to \frac{81}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-9x+\frac{81}{4}=\frac{47871}{2380}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{47871}{2380}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{47871}{2380}}$$

Simplify.

$$x-\frac{9}{2}=\frac{9\sqrt{351645}}{1190} x-\frac{9}{2}=-\frac{9\sqrt{351645}}{1190}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{9\sqrt{351645}}{1190}+\frac{9}{2} x=-\frac{9\sqrt{351645}}{1190}+\frac{9}{2}$$