Solve 333200(8.88+x)-53000*0.5*left(8.88+xright)^2-333200*8.88+53000*0.5*8.88^2=-17000*1.69

$$333200(8.88+x)-53000 \times 0.5 \times { \left(8.88+x \right) }^{ 2 } -333200 \times 8.88+53000 \times 0.5 \times { 8.88 }^{ 2 } =-17000 \times 1.69$$

Solve for x

$$x=\frac{\sqrt{54837834}}{2650}-\frac{3436}{1325}\approx 0.201229127$$

$$x=-\frac{\sqrt{54837834}}{2650}-\frac{3436}{1325}\approx -5.387644221$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 333200 by 8.88+x.

$$2958816+333200x-53000\times 0.5\left(8.88+x\right)^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Multiply 53000 and 0.5 to get 26500.

$$2958816+333200x-26500\left(8.88+x\right)^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8.88+x\right)^{2}.

$$2958816+333200x-26500\left(78.8544+17.76x+x^{2}\right)-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Use the distributive property to multiply -26500 by 78.8544+17.76x+x^{2}.

$$2958816+333200x-2089641.6-470640x-26500x^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Subtract 2089641.6 from 2958816 to get 869174.4.

$$869174.4+333200x-470640x-26500x^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Combine 333200x and -470640x to get -137440x.

$$869174.4-137440x-26500x^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Multiply 333200 and 8.88 to get 2958816.

$$869174.4-137440x-26500x^{2}-2958816+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Subtract 2958816 from 869174.4 to get -2089641.6.

$$-2089641.6-137440x-26500x^{2}+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Multiply 53000 and 0.5 to get 26500.

$$-2089641.6-137440x-26500x^{2}+26500\times 8.88^{2}=-17000\times 1.69$$

Calculate 8.88 to the power of 2 and get 78.8544.

$$-2089641.6-137440x-26500x^{2}+26500\times 78.8544=-17000\times 1.69$$

Multiply 26500 and 78.8544 to get 2089641.6.

$$-2089641.6-137440x-26500x^{2}+2089641.6=-17000\times 1.69$$

Add -2089641.6 and 2089641.6 to get 0.

$$-137440x-26500x^{2}=-17000\times 1.69$$

Multiply -17000 and 1.69 to get -28730.

$$-137440x-26500x^{2}=-28730$$

Add 28730 to both sides.

$$-137440x-26500x^{2}+28730=0$$

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.

$$-26500x^{2}-137440x+28730=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -26500 for a, -137440 for b, and 28730 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-137440\right)±\sqrt{\left(-137440\right)^{2}-4\left(-26500\right)\times 28730}}{2\left(-26500\right)}$$

Square -137440.

$$x=\frac{-\left(-137440\right)±\sqrt{18889753600-4\left(-26500\right)\times 28730}}{2\left(-26500\right)}$$

Multiply -4 times -26500.

$$x=\frac{-\left(-137440\right)±\sqrt{18889753600+106000\times 28730}}{2\left(-26500\right)}$$

Multiply 106000 times 28730.

$$x=\frac{-\left(-137440\right)±\sqrt{18889753600+3045380000}}{2\left(-26500\right)}$$

Add 18889753600 to 3045380000.

$$x=\frac{-\left(-137440\right)±\sqrt{21935133600}}{2\left(-26500\right)}$$

Take the square root of 21935133600.

$$x=\frac{-\left(-137440\right)±20\sqrt{54837834}}{2\left(-26500\right)}$$

The opposite of -137440 is 137440.

$$x=\frac{137440±20\sqrt{54837834}}{2\left(-26500\right)}$$

Multiply 2 times -26500.

$$x=\frac{137440±20\sqrt{54837834}}{-53000}$$

Now solve the equation x=\frac{137440±20\sqrt{54837834}}{-53000} when ± is plus. Add 137440 to 20\sqrt{54837834}.

$$x=\frac{20\sqrt{54837834}+137440}{-53000}$$

Divide 137440+20\sqrt{54837834} by -53000.

$$x=-\frac{\sqrt{54837834}}{2650}-\frac{3436}{1325}$$

Now solve the equation x=\frac{137440±20\sqrt{54837834}}{-53000} when ± is minus. Subtract 20\sqrt{54837834} from 137440.

$$x=\frac{137440-20\sqrt{54837834}}{-53000}$$

Divide 137440-20\sqrt{54837834} by -53000.

$$x=\frac{\sqrt{54837834}}{2650}-\frac{3436}{1325}$$

The equation is now solved.

$$x=-\frac{\sqrt{54837834}}{2650}-\frac{3436}{1325} x=\frac{\sqrt{54837834}}{2650}-\frac{3436}{1325}$$

Steps for Completing the Square

Use the distributive property to multiply 333200 by 8.88+x.

$$2958816+333200x-53000\times 0.5\left(8.88+x\right)^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Multiply 53000 and 0.5 to get 26500.

$$2958816+333200x-26500\left(8.88+x\right)^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8.88+x\right)^{2}.

$$2958816+333200x-26500\left(78.8544+17.76x+x^{2}\right)-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Use the distributive property to multiply -26500 by 78.8544+17.76x+x^{2}.

$$2958816+333200x-2089641.6-470640x-26500x^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Subtract 2089641.6 from 2958816 to get 869174.4.

$$869174.4+333200x-470640x-26500x^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Combine 333200x and -470640x to get -137440x.

$$869174.4-137440x-26500x^{2}-333200\times 8.88+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Multiply 333200 and 8.88 to get 2958816.

$$869174.4-137440x-26500x^{2}-2958816+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Subtract 2958816 from 869174.4 to get -2089641.6.

$$-2089641.6-137440x-26500x^{2}+53000\times 0.5\times 8.88^{2}=-17000\times 1.69$$

Multiply 53000 and 0.5 to get 26500.

$$-2089641.6-137440x-26500x^{2}+26500\times 8.88^{2}=-17000\times 1.69$$

Calculate 8.88 to the power of 2 and get 78.8544.

$$-2089641.6-137440x-26500x^{2}+26500\times 78.8544=-17000\times 1.69$$

Multiply 26500 and 78.8544 to get 2089641.6.

$$-2089641.6-137440x-26500x^{2}+2089641.6=-17000\times 1.69$$

Add -2089641.6 and 2089641.6 to get 0.

$$-137440x-26500x^{2}=-17000\times 1.69$$

Multiply -17000 and 1.69 to get -28730.

$$-137440x-26500x^{2}=-28730$$

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.

$$-26500x^{2}-137440x=-28730$$

Divide both sides by -26500.

$$\frac{-26500x^{2}-137440x}{-26500}=-\frac{28730}{-26500}$$

Dividing by -26500 undoes the multiplication by -26500.

$$x^{2}+\left(-\frac{137440}{-26500}\right)x=-\frac{28730}{-26500}$$

Reduce the fraction \frac{-137440}{-26500} to lowest terms by extracting and canceling out 20.

$$x^{2}+\frac{6872}{1325}x=-\frac{28730}{-26500}$$

Reduce the fraction \frac{-28730}{-26500} to lowest terms by extracting and canceling out 10.

$$x^{2}+\frac{6872}{1325}x=\frac{2873}{2650}$$

Divide \frac{6872}{1325}, the coefficient of the x term, by 2 to get \frac{3436}{1325}. Then add the square of \frac{3436}{1325} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{6872}{1325}x+\left(\frac{3436}{1325}\right)^{2}=\frac{2873}{2650}+\left(\frac{3436}{1325}\right)^{2}$$

Square \frac{3436}{1325} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6872}{1325}x+\frac{11806096}{1755625}=\frac{2873}{2650}+\frac{11806096}{1755625}$$

Add \frac{2873}{2650} to \frac{11806096}{1755625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6872}{1325}x+\frac{11806096}{1755625}=\frac{27418917}{3511250}$$

Factor x^{2}+\frac{6872}{1325}x+\frac{11806096}{1755625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3436}{1325}\right)^{2}=\frac{27418917}{3511250}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3436}{1325}\right)^{2}}=\sqrt{\frac{27418917}{3511250}}$$

Simplify.

$$x+\frac{3436}{1325}=\frac{\sqrt{54837834}}{2650} x+\frac{3436}{1325}=-\frac{\sqrt{54837834}}{2650}$$

Subtract \frac{3436}{1325} from both sides of the equation.

$$x=\frac{\sqrt{54837834}}{2650}-\frac{3436}{1325} x=-\frac{\sqrt{54837834}}{2650}-\frac{3436}{1325}$$