Solve (13.09x-12.7)*80=165.5*41+13.09x(47.5-0.4x)+178.2*41

$$(13.09x-12.7) \times 80=165.5 \times 41+13.09x(47.5-0.4x)+178.2 \times 41$$

Solve for x

$$x=\frac{5\sqrt{19896083977}}{10472}-40.625\approx 26.722917847$$

$$x=-\frac{5\sqrt{19896083977}}{10472}-40.625\approx -107.972917847$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 13.09x-12.7 by 80.

$$1047.2x-1016=165.5\times 41+13.09x\left(47.5-0.4x\right)+178.2\times 41$$

Multiply 165.5 and 41 to get 6785.5.

$$1047.2x-1016=6785.5+13.09x\left(47.5-0.4x\right)+178.2\times 41$$

Use the distributive property to multiply 13.09x by 47.5-0.4x.

$$1047.2x-1016=6785.5+621.775x-5.236x^{2}+178.2\times 41$$

Multiply 178.2 and 41 to get 7306.2.

$$1047.2x-1016=6785.5+621.775x-5.236x^{2}+7306.2$$

Add 6785.5 and 7306.2 to get 14091.7.

$$1047.2x-1016=14091.7+621.775x-5.236x^{2}$$

Subtract 14091.7 from both sides.

$$1047.2x-1016-14091.7=621.775x-5.236x^{2}$$

Subtract 14091.7 from -1016 to get -15107.7.

$$1047.2x-15107.7=621.775x-5.236x^{2}$$

Subtract 621.775x from both sides.

$$1047.2x-15107.7-621.775x=-5.236x^{2}$$

Combine 1047.2x and -621.775x to get 425.425x.

$$425.425x-15107.7=-5.236x^{2}$$

Add 5.236x^{2} to both sides.

$$425.425x-15107.7+5.236x^{2}=0$$

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.

$$5.236x^{2}+425.425x-15107.7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5.236 for a, 425.425 for b, and -15107.7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-425.425±\sqrt{425.425^{2}-4\times 5.236\left(-15107.7\right)}}{2\times 5.236}$$

Square 425.425 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-425.425±\sqrt{180986.430625-4\times 5.236\left(-15107.7\right)}}{2\times 5.236}$$

Multiply -4 times 5.236.

$$x=\frac{-425.425±\sqrt{180986.430625-20.944\left(-15107.7\right)}}{2\times 5.236}$$

Multiply -20.944 times -15107.7 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-425.425±\sqrt{180986.430625+316415.6688}}{2\times 5.236}$$

Add 180986.430625 to 316415.6688 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-425.425±\sqrt{497402.099425}}{2\times 5.236}$$

Take the square root of 497402.099425.

$$x=\frac{-425.425±\frac{\sqrt{19896083977}}{200}}{2\times 5.236}$$

Multiply 2 times 5.236.

$$x=\frac{-425.425±\frac{\sqrt{19896083977}}{200}}{10.472}$$

Now solve the equation x=\frac{-425.425±\frac{\sqrt{19896083977}}{200}}{10.472} when ± is plus. Add -425.425 to \frac{\sqrt{19896083977}}{200}.

$$x=\frac{\frac{\sqrt{19896083977}}{200}-\frac{17017}{40}}{10.472}$$

Divide -\frac{17017}{40}+\frac{\sqrt{19896083977}}{200} by 10.472 by multiplying -\frac{17017}{40}+\frac{\sqrt{19896083977}}{200} by the reciprocal of 10.472.

$$x=\frac{5\sqrt{19896083977}}{10472}-\frac{325}{8}$$

Now solve the equation x=\frac{-425.425±\frac{\sqrt{19896083977}}{200}}{10.472} when ± is minus. Subtract \frac{\sqrt{19896083977}}{200} from -425.425.

$$x=\frac{-\frac{\sqrt{19896083977}}{200}-\frac{17017}{40}}{10.472}$$

Divide -\frac{17017}{40}-\frac{\sqrt{19896083977}}{200} by 10.472 by multiplying -\frac{17017}{40}-\frac{\sqrt{19896083977}}{200} by the reciprocal of 10.472.

$$x=-\frac{5\sqrt{19896083977}}{10472}-\frac{325}{8}$$

The equation is now solved.

$$x=\frac{5\sqrt{19896083977}}{10472}-\frac{325}{8} x=-\frac{5\sqrt{19896083977}}{10472}-\frac{325}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 13.09x-12.7 by 80.

$$1047.2x-1016=165.5\times 41+13.09x\left(47.5-0.4x\right)+178.2\times 41$$

Multiply 165.5 and 41 to get 6785.5.

$$1047.2x-1016=6785.5+13.09x\left(47.5-0.4x\right)+178.2\times 41$$

Use the distributive property to multiply 13.09x by 47.5-0.4x.

$$1047.2x-1016=6785.5+621.775x-5.236x^{2}+178.2\times 41$$

Multiply 178.2 and 41 to get 7306.2.

$$1047.2x-1016=6785.5+621.775x-5.236x^{2}+7306.2$$

Add 6785.5 and 7306.2 to get 14091.7.

$$1047.2x-1016=14091.7+621.775x-5.236x^{2}$$

Subtract 621.775x from both sides.

$$1047.2x-1016-621.775x=14091.7-5.236x^{2}$$

Combine 1047.2x and -621.775x to get 425.425x.

$$425.425x-1016=14091.7-5.236x^{2}$$

Add 5.236x^{2} to both sides.

$$425.425x-1016+5.236x^{2}=14091.7$$

Add 1016 to both sides.

$$425.425x+5.236x^{2}=14091.7+1016$$

Add 14091.7 and 1016 to get 15107.7.

$$425.425x+5.236x^{2}=15107.7$$

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.

$$5.236x^{2}+425.425x=15107.7$$

Divide both sides of the equation by 5.236, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{5.236x^{2}+425.425x}{5.236}=\frac{15107.7}{5.236}$$

Dividing by 5.236 undoes the multiplication by 5.236.

$$x^{2}+\frac{425.425}{5.236}x=\frac{15107.7}{5.236}$$

Divide 425.425 by 5.236 by multiplying 425.425 by the reciprocal of 5.236.

$$x^{2}+81.25x=\frac{15107.7}{5.236}$$

Divide 15107.7 by 5.236 by multiplying 15107.7 by the reciprocal of 5.236.

$$x^{2}+81.25x=\frac{3776925}{1309}$$

Divide 81.25, the coefficient of the x term, by 2 to get 40.625. Then add the square of 40.625 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+81.25x+40.625^{2}=\frac{3776925}{1309}+40.625^{2}$$

Square 40.625 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+81.25x+1650.390625=\frac{3776925}{1309}+1650.390625$$

Add \frac{3776925}{1309} to 1650.390625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+81.25x+1650.390625=\frac{379986325}{83776}$$

Factor x^{2}+81.25x+1650.390625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+40.625\right)^{2}=\frac{379986325}{83776}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+40.625\right)^{2}}=\sqrt{\frac{379986325}{83776}}$$

Simplify.

$$x+40.625=\frac{5\sqrt{19896083977}}{10472} x+40.625=-\frac{5\sqrt{19896083977}}{10472}$$

Subtract 40.625 from both sides of the equation.

$$x=\frac{5\sqrt{19896083977}}{10472}-\frac{325}{8} x=-\frac{5\sqrt{19896083977}}{10472}-\frac{325}{8}$$