Solve for x
x = \frac{5 \sqrt{91} + 48}{29} \approx 3.299895175
x=\frac{48-5\sqrt{91}}{29}\approx 0.010449653
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256\left(x^{2}\right)^{2}-384x^{2}x+144x^{2}=4\left(1+x^{2}\right)\left(64x^{2}-96x+1\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16x^{2}-12x\right)^{2}.
256x^{4}-384x^{2}x+144x^{2}=4\left(1+x^{2}\right)\left(64x^{2}-96x+1\right)
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
256x^{4}-384x^{3}+144x^{2}=4\left(1+x^{2}\right)\left(64x^{2}-96x+1\right)
To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.
256x^{4}-384x^{3}+144x^{2}=\left(4+4x^{2}\right)\left(64x^{2}-96x+1\right)
Use the distributive property to multiply 4 by 1+x^{2}.
256x^{4}-384x^{3}+144x^{2}=260x^{2}-384x+4+256x^{4}-384x^{3}
Use the distributive property to multiply 4+4x^{2} by 64x^{2}-96x+1 and combine like terms.
256x^{4}-384x^{3}+144x^{2}-260x^{2}=-384x+4+256x^{4}-384x^{3}
Subtract 260x^{2} from both sides.
256x^{4}-384x^{3}-116x^{2}=-384x+4+256x^{4}-384x^{3}
Combine 144x^{2} and -260x^{2} to get -116x^{2}.
256x^{4}-384x^{3}-116x^{2}+384x=4+256x^{4}-384x^{3}
Add 384x to both sides.
256x^{4}-384x^{3}-116x^{2}+384x-4=256x^{4}-384x^{3}
Subtract 4 from both sides.
256x^{4}-384x^{3}-116x^{2}+384x-4-256x^{4}=-384x^{3}
Subtract 256x^{4} from both sides.
-384x^{3}-116x^{2}+384x-4=-384x^{3}
Combine 256x^{4} and -256x^{4} to get 0.
-384x^{3}-116x^{2}+384x-4+384x^{3}=0
Add 384x^{3} to both sides.
-116x^{2}+384x-4=0
Combine -384x^{3} and 384x^{3} to get 0.
x=\frac{-384±\sqrt{384^{2}-4\left(-116\right)\left(-4\right)}}{2\left(-116\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -116 for a, 384 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-384±\sqrt{147456-4\left(-116\right)\left(-4\right)}}{2\left(-116\right)}
Square 384.
x=\frac{-384±\sqrt{147456+464\left(-4\right)}}{2\left(-116\right)}
Multiply -4 times -116.
x=\frac{-384±\sqrt{147456-1856}}{2\left(-116\right)}
Multiply 464 times -4.
x=\frac{-384±\sqrt{145600}}{2\left(-116\right)}
Add 147456 to -1856.
x=\frac{-384±40\sqrt{91}}{2\left(-116\right)}
Take the square root of 145600.
x=\frac{-384±40\sqrt{91}}{-232}
Multiply 2 times -116.
x=\frac{40\sqrt{91}-384}{-232}
Now solve the equation x=\frac{-384±40\sqrt{91}}{-232} when ± is plus. Add -384 to 40\sqrt{91}.
x=\frac{48-5\sqrt{91}}{29}
Divide -384+40\sqrt{91} by -232.
x=\frac{-40\sqrt{91}-384}{-232}
Now solve the equation x=\frac{-384±40\sqrt{91}}{-232} when ± is minus. Subtract 40\sqrt{91} from -384.
x=\frac{5\sqrt{91}+48}{29}
Divide -384-40\sqrt{91} by -232.
x=\frac{48-5\sqrt{91}}{29} x=\frac{5\sqrt{91}+48}{29}
The equation is now solved.
256\left(x^{2}\right)^{2}-384x^{2}x+144x^{2}=4\left(1+x^{2}\right)\left(64x^{2}-96x+1\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16x^{2}-12x\right)^{2}.
256x^{4}-384x^{2}x+144x^{2}=4\left(1+x^{2}\right)\left(64x^{2}-96x+1\right)
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
256x^{4}-384x^{3}+144x^{2}=4\left(1+x^{2}\right)\left(64x^{2}-96x+1\right)
To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.
256x^{4}-384x^{3}+144x^{2}=\left(4+4x^{2}\right)\left(64x^{2}-96x+1\right)
Use the distributive property to multiply 4 by 1+x^{2}.
256x^{4}-384x^{3}+144x^{2}=260x^{2}-384x+4+256x^{4}-384x^{3}
Use the distributive property to multiply 4+4x^{2} by 64x^{2}-96x+1 and combine like terms.
256x^{4}-384x^{3}+144x^{2}-260x^{2}=-384x+4+256x^{4}-384x^{3}
Subtract 260x^{2} from both sides.
256x^{4}-384x^{3}-116x^{2}=-384x+4+256x^{4}-384x^{3}
Combine 144x^{2} and -260x^{2} to get -116x^{2}.
256x^{4}-384x^{3}-116x^{2}+384x=4+256x^{4}-384x^{3}
Add 384x to both sides.
256x^{4}-384x^{3}-116x^{2}+384x-256x^{4}=4-384x^{3}
Subtract 256x^{4} from both sides.
-384x^{3}-116x^{2}+384x=4-384x^{3}
Combine 256x^{4} and -256x^{4} to get 0.
-384x^{3}-116x^{2}+384x+384x^{3}=4
Add 384x^{3} to both sides.
-116x^{2}+384x=4
Combine -384x^{3} and 384x^{3} to get 0.
\frac{-116x^{2}+384x}{-116}=\frac{4}{-116}
Divide both sides by -116.
x^{2}+\frac{384}{-116}x=\frac{4}{-116}
Dividing by -116 undoes the multiplication by -116.
x^{2}-\frac{96}{29}x=\frac{4}{-116}
Reduce the fraction \frac{384}{-116} to lowest terms by extracting and canceling out 4.
x^{2}-\frac{96}{29}x=-\frac{1}{29}
Reduce the fraction \frac{4}{-116} to lowest terms by extracting and canceling out 4.
x^{2}-\frac{96}{29}x+\left(-\frac{48}{29}\right)^{2}=-\frac{1}{29}+\left(-\frac{48}{29}\right)^{2}
Divide -\frac{96}{29}, the coefficient of the x term, by 2 to get -\frac{48}{29}. Then add the square of -\frac{48}{29} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{96}{29}x+\frac{2304}{841}=-\frac{1}{29}+\frac{2304}{841}
Square -\frac{48}{29} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{96}{29}x+\frac{2304}{841}=\frac{2275}{841}
Add -\frac{1}{29} to \frac{2304}{841} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{48}{29}\right)^{2}=\frac{2275}{841}
Factor x^{2}-\frac{96}{29}x+\frac{2304}{841}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{48}{29}\right)^{2}}=\sqrt{\frac{2275}{841}}
Take the square root of both sides of the equation.
x-\frac{48}{29}=\frac{5\sqrt{91}}{29} x-\frac{48}{29}=-\frac{5\sqrt{91}}{29}
Simplify.
x=\frac{5\sqrt{91}+48}{29} x=\frac{48-5\sqrt{91}}{29}
Add \frac{48}{29} to both sides of the equation.
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Limits
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