Solve 2*75(1936+4x^2-176x)+81*5(176*4x^2)=4204

$$2 \cdot 75 ( 1936 + 4 x ^ { 2 } - 176 x ) + 81 \cdot 5 ( 176 \cdot 4 x ^ { 2 } ) = 4204$$

Solve for x (complex solution)

$$x=\frac{\sqrt{5099855070}i}{71430}+\frac{110}{2381}\approx 0.046199076+0.999765779i$$

$$x=-\frac{\sqrt{5099855070}i}{71430}+\frac{110}{2381}\approx 0.046199076-0.999765779i$$

Steps Using the Quadratic Formula

Do the multiplications.

$$150\left(1936+4x^{2}-176x\right)+405\times 176\times 4x^{2}=4204$$

Use the distributive property to multiply 150 by 1936+4x^{2}-176x.

$$290400+600x^{2}-26400x+405\times 176\times 4x^{2}=4204$$

Multiply 405 and 176 to get 71280.

$$290400+600x^{2}-26400x+71280\times 4x^{2}=4204$$

Multiply 71280 and 4 to get 285120.

$$290400+600x^{2}-26400x+285120x^{2}=4204$$

Combine 600x^{2} and 285120x^{2} to get 285720x^{2}.

$$290400+285720x^{2}-26400x=4204$$

Subtract 4204 from both sides.

$$290400+285720x^{2}-26400x-4204=0$$

Subtract 4204 from 290400 to get 286196.

$$286196+285720x^{2}-26400x=0$$

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.

$$285720x^{2}-26400x+286196=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 285720 for a, -26400 for b, and 286196 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26400\right)±\sqrt{\left(-26400\right)^{2}-4\times 285720\times 286196}}{2\times 285720}$$

Square -26400.

$$x=\frac{-\left(-26400\right)±\sqrt{696960000-4\times 285720\times 286196}}{2\times 285720}$$

Multiply -4 times 285720.

$$x=\frac{-\left(-26400\right)±\sqrt{696960000-1142880\times 286196}}{2\times 285720}$$

Multiply -1142880 times 286196.

$$x=\frac{-\left(-26400\right)±\sqrt{696960000-327087684480}}{2\times 285720}$$

Add 696960000 to -327087684480.

$$x=\frac{-\left(-26400\right)±\sqrt{-326390724480}}{2\times 285720}$$

Take the square root of -326390724480.

$$x=\frac{-\left(-26400\right)±8\sqrt{5099855070}i}{2\times 285720}$$

The opposite of -26400 is 26400.

$$x=\frac{26400±8\sqrt{5099855070}i}{2\times 285720}$$

Multiply 2 times 285720.

$$x=\frac{26400±8\sqrt{5099855070}i}{571440}$$

Now solve the equation x=\frac{26400±8\sqrt{5099855070}i}{571440} when ± is plus. Add 26400 to 8i\sqrt{5099855070}.

$$x=\frac{26400+8\sqrt{5099855070}i}{571440}$$

Divide 26400+8i\sqrt{5099855070} by 571440.

$$x=\frac{\sqrt{5099855070}i}{71430}+\frac{110}{2381}$$

Now solve the equation x=\frac{26400±8\sqrt{5099855070}i}{571440} when ± is minus. Subtract 8i\sqrt{5099855070} from 26400.

$$x=\frac{-8\sqrt{5099855070}i+26400}{571440}$$

Divide 26400-8i\sqrt{5099855070} by 571440.

$$x=-\frac{\sqrt{5099855070}i}{71430}+\frac{110}{2381}$$

The equation is now solved.

$$x=\frac{\sqrt{5099855070}i}{71430}+\frac{110}{2381} x=-\frac{\sqrt{5099855070}i}{71430}+\frac{110}{2381}$$

Steps for Completing the Square

Do the multiplications.

$$150\left(1936+4x^{2}-176x\right)+405\times 176\times 4x^{2}=4204$$

Use the distributive property to multiply 150 by 1936+4x^{2}-176x.

$$290400+600x^{2}-26400x+405\times 176\times 4x^{2}=4204$$

Multiply 405 and 176 to get 71280.

$$290400+600x^{2}-26400x+71280\times 4x^{2}=4204$$

Multiply 71280 and 4 to get 285120.

$$290400+600x^{2}-26400x+285120x^{2}=4204$$

Combine 600x^{2} and 285120x^{2} to get 285720x^{2}.

$$290400+285720x^{2}-26400x=4204$$

Subtract 290400 from both sides.

$$285720x^{2}-26400x=4204-290400$$

Subtract 290400 from 4204 to get -286196.

$$285720x^{2}-26400x=-286196$$

Divide both sides by 285720.

$$\frac{285720x^{2}-26400x}{285720}=-\frac{286196}{285720}$$

Dividing by 285720 undoes the multiplication by 285720.

$$x^{2}+\left(-\frac{26400}{285720}\right)x=-\frac{286196}{285720}$$

Reduce the fraction \frac{-26400}{285720} to lowest terms by extracting and canceling out 120.

$$x^{2}-\frac{220}{2381}x=-\frac{286196}{285720}$$

Reduce the fraction \frac{-286196}{285720} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{220}{2381}x=-\frac{71549}{71430}$$

Divide -\frac{220}{2381}, the coefficient of the x term, by 2 to get -\frac{110}{2381}. Then add the square of -\frac{110}{2381} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{220}{2381}x+\left(-\frac{110}{2381}\right)^{2}=-\frac{71549}{71430}+\left(-\frac{110}{2381}\right)^{2}$$

Square -\frac{110}{2381} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{220}{2381}x+\frac{12100}{5669161}=-\frac{71549}{71430}+\frac{12100}{5669161}$$

Add -\frac{71549}{71430} to \frac{12100}{5669161} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{220}{2381}x+\frac{12100}{5669161}=-\frac{169995169}{170074830}$$

Factor x^{2}-\frac{220}{2381}x+\frac{12100}{5669161}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{110}{2381}\right)^{2}=-\frac{169995169}{170074830}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{110}{2381}\right)^{2}}=\sqrt{-\frac{169995169}{170074830}}$$

Simplify.

$$x-\frac{110}{2381}=\frac{\sqrt{5099855070}i}{71430} x-\frac{110}{2381}=-\frac{\sqrt{5099855070}i}{71430}$$

Add \frac{110}{2381} to both sides of the equation.

$$x=\frac{\sqrt{5099855070}i}{71430}+\frac{110}{2381} x=-\frac{\sqrt{5099855070}i}{71430}+\frac{110}{2381}$$