Solve for x (complex solution)
x=\frac{58+4\sqrt{26}i}{7}\approx 8.285714286+2.913725436i
x=\frac{-4\sqrt{26}i+58}{7}\approx 8.285714286-2.913725436i
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42x^{2}-696x+3240=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.
x=\frac{-\left(-696\right)±\sqrt{\left(-696\right)^{2}-4\times 42\times 3240}}{2\times 42}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 42 for a, -696 for b, and 3240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-696\right)±\sqrt{484416-4\times 42\times 3240}}{2\times 42}
Square -696.
x=\frac{-\left(-696\right)±\sqrt{484416-168\times 3240}}{2\times 42}
Multiply -4 times 42.
x=\frac{-\left(-696\right)±\sqrt{484416-544320}}{2\times 42}
Multiply -168 times 3240.
x=\frac{-\left(-696\right)±\sqrt{-59904}}{2\times 42}
Add 484416 to -544320.
x=\frac{-\left(-696\right)±48\sqrt{26}i}{2\times 42}
Take the square root of -59904.
x=\frac{696±48\sqrt{26}i}{2\times 42}
The opposite of -696 is 696.
x=\frac{696±48\sqrt{26}i}{84}
Multiply 2 times 42.
x=\frac{696+48\sqrt{26}i}{84}
Now solve the equation x=\frac{696±48\sqrt{26}i}{84} when ± is plus. Add 696 to 48i\sqrt{26}.
x=\frac{58+4\sqrt{26}i}{7}
Divide 696+48i\sqrt{26} by 84.
x=\frac{-48\sqrt{26}i+696}{84}
Now solve the equation x=\frac{696±48\sqrt{26}i}{84} when ± is minus. Subtract 48i\sqrt{26} from 696.
x=\frac{-4\sqrt{26}i+58}{7}
Divide 696-48i\sqrt{26} by 84.
x=\frac{58+4\sqrt{26}i}{7} x=\frac{-4\sqrt{26}i+58}{7}
The equation is now solved.
42x^{2}-696x+3240=0
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.
42x^{2}-696x+3240-3240=-3240
Subtract 3240 from both sides of the equation.
42x^{2}-696x=-3240
Subtracting 3240 from itself leaves 0.
\frac{42x^{2}-696x}{42}=-\frac{3240}{42}
Divide both sides by 42.
x^{2}+\left(-\frac{696}{42}\right)x=-\frac{3240}{42}
Dividing by 42 undoes the multiplication by 42.
x^{2}-\frac{116}{7}x=-\frac{3240}{42}
Reduce the fraction \frac{-696}{42} to lowest terms by extracting and canceling out 6.
x^{2}-\frac{116}{7}x=-\frac{540}{7}
Reduce the fraction \frac{-3240}{42} to lowest terms by extracting and canceling out 6.
x^{2}-\frac{116}{7}x+\left(-\frac{58}{7}\right)^{2}=-\frac{540}{7}+\left(-\frac{58}{7}\right)^{2}
Divide -\frac{116}{7}, the coefficient of the x term, by 2 to get -\frac{58}{7}. Then add the square of -\frac{58}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{116}{7}x+\frac{3364}{49}=-\frac{540}{7}+\frac{3364}{49}
Square -\frac{58}{7} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{116}{7}x+\frac{3364}{49}=-\frac{416}{49}
Add -\frac{540}{7} to \frac{3364}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{58}{7}\right)^{2}=-\frac{416}{49}
Factor x^{2}-\frac{116}{7}x+\frac{3364}{49}. 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{58}{7}\right)^{2}}=\sqrt{-\frac{416}{49}}
Take the square root of both sides of the equation.
x-\frac{58}{7}=\frac{4\sqrt{26}i}{7} x-\frac{58}{7}=-\frac{4\sqrt{26}i}{7}
Simplify.
x=\frac{58+4\sqrt{26}i}{7} x=\frac{-4\sqrt{26}i+58}{7}
Add \frac{58}{7} to both sides of the equation.
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Simultaneous equation
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Differentiation
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Limits
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