Solve 7.5=-1/0.04left(left(1-0.04xright)^2/0.7-1/0.7right)

$$7.5 = \frac{ -1 }{ 0.04 } \left( \frac{ { \left(1-0.04x \right) }^{ 2 } }{ 0.7 } - \frac{ 1 }{ 0.7 } \right)$$

Solve for x

$$x=\frac{5\sqrt{79}}{2}+25\approx 47.220486043$$

$$x=-\frac{5\sqrt{79}}{2}+25\approx 2.779513957$$

Steps Using the Quadratic Formula

Expand \frac{-1}{0.04} by multiplying both numerator and the denominator by 100.

$$7.5=\frac{-100}{4}\left(\frac{\left(1-0.04x\right)^{2}}{0.7}-\frac{1}{0.7}\right)$$

Divide -100 by 4 to get -25.

$$7.5=-25\left(\frac{\left(1-0.04x\right)^{2}}{0.7}-\frac{1}{0.7}\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-0.04x\right)^{2}.

$$7.5=-25\left(\frac{1-0.08x+0.0016x^{2}}{0.7}-\frac{1}{0.7}\right)$$

Expand \frac{1}{0.7} by multiplying both numerator and the denominator by 10.

$$7.5=-25\left(\frac{1-0.08x+0.0016x^{2}}{0.7}-\frac{10}{7}\right)$$

Use the distributive property to multiply -25 by \frac{1-0.08x+0.0016x^{2}}{0.7}-\frac{10}{7}.

$$7.5=-25\times \frac{1-0.08x+0.0016x^{2}}{0.7}+\frac{250}{7}$$

Divide each term of 1-0.08x+0.0016x^{2} by 0.7 to get \frac{1}{0.7}+\frac{-0.08x}{0.7}+\frac{0.0016x^{2}}{0.7}.

$$7.5=-25\left(\frac{1}{0.7}+\frac{-0.08x}{0.7}+\frac{0.0016x^{2}}{0.7}\right)+\frac{250}{7}$$

Expand \frac{1}{0.7} by multiplying both numerator and the denominator by 10.

$$7.5=-25\left(\frac{10}{7}+\frac{-0.08x}{0.7}+\frac{0.0016x^{2}}{0.7}\right)+\frac{250}{7}$$

Divide -0.08x by 0.7 to get -\frac{4}{35}x.

$$7.5=-25\left(\frac{10}{7}-\frac{4}{35}x+\frac{0.0016x^{2}}{0.7}\right)+\frac{250}{7}$$

Divide 0.0016x^{2} by 0.7 to get \frac{2}{875}x^{2}.

$$7.5=-25\left(\frac{10}{7}-\frac{4}{35}x+\frac{2}{875}x^{2}\right)+\frac{250}{7}$$

Use the distributive property to multiply -25 by \frac{10}{7}-\frac{4}{35}x+\frac{2}{875}x^{2}.

$$7.5=-\frac{250}{7}+\frac{20}{7}x-\frac{2}{35}x^{2}+\frac{250}{7}$$

Add -\frac{250}{7} and \frac{250}{7} to get 0.

$$7.5=\frac{20}{7}x-\frac{2}{35}x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{20}{7}x-\frac{2}{35}x^{2}=7.5$$

Subtract 7.5 from both sides.

$$\frac{20}{7}x-\frac{2}{35}x^{2}-7.5=0$$

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.

$$-\frac{2}{35}x^{2}+\frac{20}{7}x-7.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{2}{35} for a, \frac{20}{7} for b, and -7.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{20}{7}±\sqrt{\frac{20}{7}^{2}-4\left(-\frac{2}{35}\right)\left(-7.5\right)}}{2\left(-\frac{2}{35}\right)}$$

Square \frac{20}{7} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{20}{7}±\sqrt{\frac{400}{49}-4\left(-\frac{2}{35}\right)\left(-7.5\right)}}{2\left(-\frac{2}{35}\right)}$$

Multiply -4 times -\frac{2}{35}.

$$x=\frac{-\frac{20}{7}±\sqrt{\frac{400}{49}+\frac{8}{35}\left(-7.5\right)}}{2\left(-\frac{2}{35}\right)}$$

Multiply \frac{8}{35} times -7.5 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{20}{7}±\sqrt{\frac{400}{49}-\frac{12}{7}}}{2\left(-\frac{2}{35}\right)}$$

Add \frac{400}{49} to -\frac{12}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{20}{7}±\sqrt{\frac{316}{49}}}{2\left(-\frac{2}{35}\right)}$$

Take the square root of \frac{316}{49}.

$$x=\frac{-\frac{20}{7}±\frac{2\sqrt{79}}{7}}{2\left(-\frac{2}{35}\right)}$$

Multiply 2 times -\frac{2}{35}.

$$x=\frac{-\frac{20}{7}±\frac{2\sqrt{79}}{7}}{-\frac{4}{35}}$$

Now solve the equation x=\frac{-\frac{20}{7}±\frac{2\sqrt{79}}{7}}{-\frac{4}{35}} when ± is plus. Add -\frac{20}{7} to \frac{2\sqrt{79}}{7}.

$$x=\frac{2\sqrt{79}-20}{-\frac{4}{35}\times 7}$$

Divide \frac{-20+2\sqrt{79}}{7} by -\frac{4}{35} by multiplying \frac{-20+2\sqrt{79}}{7} by the reciprocal of -\frac{4}{35}.

$$x=-\frac{5\sqrt{79}}{2}+25$$

Now solve the equation x=\frac{-\frac{20}{7}±\frac{2\sqrt{79}}{7}}{-\frac{4}{35}} when ± is minus. Subtract \frac{2\sqrt{79}}{7} from -\frac{20}{7}.

$$x=\frac{-2\sqrt{79}-20}{-\frac{4}{35}\times 7}$$

Divide \frac{-20-2\sqrt{79}}{7} by -\frac{4}{35} by multiplying \frac{-20-2\sqrt{79}}{7} by the reciprocal of -\frac{4}{35}.

$$x=\frac{5\sqrt{79}}{2}+25$$

The equation is now solved.

$$x=-\frac{5\sqrt{79}}{2}+25 x=\frac{5\sqrt{79}}{2}+25$$

Steps for Completing the Square

Expand \frac{-1}{0.04} by multiplying both numerator and the denominator by 100.

$$7.5=\frac{-100}{4}\left(\frac{\left(1-0.04x\right)^{2}}{0.7}-\frac{1}{0.7}\right)$$

Divide -100 by 4 to get -25.

$$7.5=-25\left(\frac{\left(1-0.04x\right)^{2}}{0.7}-\frac{1}{0.7}\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-0.04x\right)^{2}.

$$7.5=-25\left(\frac{1-0.08x+0.0016x^{2}}{0.7}-\frac{1}{0.7}\right)$$

Expand \frac{1}{0.7} by multiplying both numerator and the denominator by 10.

$$7.5=-25\left(\frac{1-0.08x+0.0016x^{2}}{0.7}-\frac{10}{7}\right)$$

Use the distributive property to multiply -25 by \frac{1-0.08x+0.0016x^{2}}{0.7}-\frac{10}{7}.

$$7.5=-25\times \frac{1-0.08x+0.0016x^{2}}{0.7}+\frac{250}{7}$$

Divide each term of 1-0.08x+0.0016x^{2} by 0.7 to get \frac{1}{0.7}+\frac{-0.08x}{0.7}+\frac{0.0016x^{2}}{0.7}.

$$7.5=-25\left(\frac{1}{0.7}+\frac{-0.08x}{0.7}+\frac{0.0016x^{2}}{0.7}\right)+\frac{250}{7}$$

Expand \frac{1}{0.7} by multiplying both numerator and the denominator by 10.

$$7.5=-25\left(\frac{10}{7}+\frac{-0.08x}{0.7}+\frac{0.0016x^{2}}{0.7}\right)+\frac{250}{7}$$

Divide -0.08x by 0.7 to get -\frac{4}{35}x.

$$7.5=-25\left(\frac{10}{7}-\frac{4}{35}x+\frac{0.0016x^{2}}{0.7}\right)+\frac{250}{7}$$

Divide 0.0016x^{2} by 0.7 to get \frac{2}{875}x^{2}.

$$7.5=-25\left(\frac{10}{7}-\frac{4}{35}x+\frac{2}{875}x^{2}\right)+\frac{250}{7}$$

Use the distributive property to multiply -25 by \frac{10}{7}-\frac{4}{35}x+\frac{2}{875}x^{2}.

$$7.5=-\frac{250}{7}+\frac{20}{7}x-\frac{2}{35}x^{2}+\frac{250}{7}$$

Add -\frac{250}{7} and \frac{250}{7} to get 0.

$$7.5=\frac{20}{7}x-\frac{2}{35}x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{20}{7}x-\frac{2}{35}x^{2}=7.5$$

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.

$$-\frac{2}{35}x^{2}+\frac{20}{7}x=7.5$$

Divide both sides of the equation by -\frac{2}{35}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{2}{35}x^{2}+\frac{20}{7}x}{-\frac{2}{35}}=\frac{7.5}{-\frac{2}{35}}$$

Dividing by -\frac{2}{35} undoes the multiplication by -\frac{2}{35}.

$$x^{2}+\frac{\frac{20}{7}}{-\frac{2}{35}}x=\frac{7.5}{-\frac{2}{35}}$$

Divide \frac{20}{7} by -\frac{2}{35} by multiplying \frac{20}{7} by the reciprocal of -\frac{2}{35}.

$$x^{2}-50x=\frac{7.5}{-\frac{2}{35}}$$

Divide 7.5 by -\frac{2}{35} by multiplying 7.5 by the reciprocal of -\frac{2}{35}.

$$x^{2}-50x=-131.25$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-131.25+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-131.25+625$$

Add -131.25 to 625.

$$x^{2}-50x+625=493.75$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=493.75$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{493.75}$$

Simplify.

$$x-25=\frac{5\sqrt{79}}{2} x-25=-\frac{5\sqrt{79}}{2}$$

Add 25 to both sides of the equation.

$$x=\frac{5\sqrt{79}}{2}+25 x=-\frac{5\sqrt{79}}{2}+25$$