Solve 5x+40/36x(1-7x)=40%

$$5x+ \frac{ 40 }{ 36 } x(1-7x) = 40 \%$$

Solve for x

$$x=\frac{\sqrt{2017}}{140}+\frac{11}{28}\approx 0.713650165$$

$$x=-\frac{\sqrt{2017}}{140}+\frac{11}{28}\approx 0.07206412$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{40}{36} to lowest terms by extracting and canceling out 4.

$$5x+\frac{10}{9}x\left(1-7x\right)=\frac{40}{100}$$

Use the distributive property to multiply \frac{10}{9}x by 1-7x.

$$5x+\frac{10}{9}x+\frac{10}{9}x\left(-7\right)x=\frac{40}{100}$$

Multiply x and x to get x^{2}.

$$5x+\frac{10}{9}x+\frac{10}{9}x^{2}\left(-7\right)=\frac{40}{100}$$

Express \frac{10}{9}\left(-7\right) as a single fraction.

$$5x+\frac{10}{9}x+\frac{10\left(-7\right)}{9}x^{2}=\frac{40}{100}$$

Multiply 10 and -7 to get -70.

$$5x+\frac{10}{9}x+\frac{-70}{9}x^{2}=\frac{40}{100}$$

Fraction \frac{-70}{9} can be rewritten as -\frac{70}{9} by extracting the negative sign.

$$5x+\frac{10}{9}x-\frac{70}{9}x^{2}=\frac{40}{100}$$

Combine 5x and \frac{10}{9}x to get \frac{55}{9}x.

$$\frac{55}{9}x-\frac{70}{9}x^{2}=\frac{40}{100}$$

Reduce the fraction \frac{40}{100} to lowest terms by extracting and canceling out 20.

$$\frac{55}{9}x-\frac{70}{9}x^{2}=\frac{2}{5}$$

Subtract \frac{2}{5} from both sides.

$$\frac{55}{9}x-\frac{70}{9}x^{2}-\frac{2}{5}=0$$

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.

$$-\frac{70}{9}x^{2}+\frac{55}{9}x-\frac{2}{5}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{70}{9} for a, \frac{55}{9} for b, and -\frac{2}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{55}{9}±\sqrt{\left(\frac{55}{9}\right)^{2}-4\left(-\frac{70}{9}\right)\left(-\frac{2}{5}\right)}}{2\left(-\frac{70}{9}\right)}$$

Square \frac{55}{9} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{55}{9}±\sqrt{\frac{3025}{81}-4\left(-\frac{70}{9}\right)\left(-\frac{2}{5}\right)}}{2\left(-\frac{70}{9}\right)}$$

Multiply -4 times -\frac{70}{9}.

$$x=\frac{-\frac{55}{9}±\sqrt{\frac{3025}{81}+\frac{280}{9}\left(-\frac{2}{5}\right)}}{2\left(-\frac{70}{9}\right)}$$

Multiply \frac{280}{9} times -\frac{2}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{55}{9}±\sqrt{\frac{3025}{81}-\frac{112}{9}}}{2\left(-\frac{70}{9}\right)}$$

Add \frac{3025}{81} to -\frac{112}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{55}{9}±\sqrt{\frac{2017}{81}}}{2\left(-\frac{70}{9}\right)}$$

Take the square root of \frac{2017}{81}.

$$x=\frac{-\frac{55}{9}±\frac{\sqrt{2017}}{9}}{2\left(-\frac{70}{9}\right)}$$

Multiply 2 times -\frac{70}{9}.

$$x=\frac{-\frac{55}{9}±\frac{\sqrt{2017}}{9}}{-\frac{140}{9}}$$

Now solve the equation x=\frac{-\frac{55}{9}±\frac{\sqrt{2017}}{9}}{-\frac{140}{9}} when ± is plus. Add -\frac{55}{9} to \frac{\sqrt{2017}}{9}.

$$x=\frac{\sqrt{2017}-55}{-\frac{140}{9}\times 9}$$

Divide \frac{-55+\sqrt{2017}}{9} by -\frac{140}{9} by multiplying \frac{-55+\sqrt{2017}}{9} by the reciprocal of -\frac{140}{9}.

$$x=-\frac{\sqrt{2017}}{140}+\frac{11}{28}$$

Now solve the equation x=\frac{-\frac{55}{9}±\frac{\sqrt{2017}}{9}}{-\frac{140}{9}} when ± is minus. Subtract \frac{\sqrt{2017}}{9} from -\frac{55}{9}.

$$x=\frac{-\sqrt{2017}-55}{-\frac{140}{9}\times 9}$$

Divide \frac{-55-\sqrt{2017}}{9} by -\frac{140}{9} by multiplying \frac{-55-\sqrt{2017}}{9} by the reciprocal of -\frac{140}{9}.

$$x=\frac{\sqrt{2017}}{140}+\frac{11}{28}$$

The equation is now solved.

$$x=-\frac{\sqrt{2017}}{140}+\frac{11}{28} x=\frac{\sqrt{2017}}{140}+\frac{11}{28}$$

Steps for Completing the Square

Reduce the fraction \frac{40}{36} to lowest terms by extracting and canceling out 4.

$$5x+\frac{10}{9}x\left(1-7x\right)=\frac{40}{100}$$

Use the distributive property to multiply \frac{10}{9}x by 1-7x.

$$5x+\frac{10}{9}x+\frac{10}{9}x\left(-7\right)x=\frac{40}{100}$$

Multiply x and x to get x^{2}.

$$5x+\frac{10}{9}x+\frac{10}{9}x^{2}\left(-7\right)=\frac{40}{100}$$

Express \frac{10}{9}\left(-7\right) as a single fraction.

$$5x+\frac{10}{9}x+\frac{10\left(-7\right)}{9}x^{2}=\frac{40}{100}$$

Multiply 10 and -7 to get -70.

$$5x+\frac{10}{9}x+\frac{-70}{9}x^{2}=\frac{40}{100}$$

Fraction \frac{-70}{9} can be rewritten as -\frac{70}{9} by extracting the negative sign.

$$5x+\frac{10}{9}x-\frac{70}{9}x^{2}=\frac{40}{100}$$

Combine 5x and \frac{10}{9}x to get \frac{55}{9}x.

$$\frac{55}{9}x-\frac{70}{9}x^{2}=\frac{40}{100}$$

Reduce the fraction \frac{40}{100} to lowest terms by extracting and canceling out 20.

$$\frac{55}{9}x-\frac{70}{9}x^{2}=\frac{2}{5}$$

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.

$$-\frac{70}{9}x^{2}+\frac{55}{9}x=\frac{2}{5}$$

Divide both sides of the equation by -\frac{70}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{70}{9}x^{2}+\frac{55}{9}x}{-\frac{70}{9}}=\frac{\frac{2}{5}}{-\frac{70}{9}}$$

Dividing by -\frac{70}{9} undoes the multiplication by -\frac{70}{9}.

$$x^{2}+\frac{\frac{55}{9}}{-\frac{70}{9}}x=\frac{\frac{2}{5}}{-\frac{70}{9}}$$

Divide \frac{55}{9} by -\frac{70}{9} by multiplying \frac{55}{9} by the reciprocal of -\frac{70}{9}.

$$x^{2}-\frac{11}{14}x=\frac{\frac{2}{5}}{-\frac{70}{9}}$$

Divide \frac{2}{5} by -\frac{70}{9} by multiplying \frac{2}{5} by the reciprocal of -\frac{70}{9}.

$$x^{2}-\frac{11}{14}x=-\frac{9}{175}$$

Divide -\frac{11}{14}, the coefficient of the x term, by 2 to get -\frac{11}{28}. Then add the square of -\frac{11}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{14}x+\left(-\frac{11}{28}\right)^{2}=-\frac{9}{175}+\left(-\frac{11}{28}\right)^{2}$$

Square -\frac{11}{28} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=-\frac{9}{175}+\frac{121}{784}$$

Add -\frac{9}{175} to \frac{121}{784} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=\frac{2017}{19600}$$

Factor x^{2}-\frac{11}{14}x+\frac{121}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{28}\right)^{2}=\frac{2017}{19600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{28}\right)^{2}}=\sqrt{\frac{2017}{19600}}$$

Simplify.

$$x-\frac{11}{28}=\frac{\sqrt{2017}}{140} x-\frac{11}{28}=-\frac{\sqrt{2017}}{140}$$

Add \frac{11}{28} to both sides of the equation.

$$x=\frac{\sqrt{2017}}{140}+\frac{11}{28} x=-\frac{\sqrt{2017}}{140}+\frac{11}{28}$$