Solve for c
c = \frac{8 \sqrt{6} - 3}{5} \approx 3.319183588
c=\frac{-8\sqrt{6}-3}{5}\approx -4.519183588
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-6c=5\left(c^{2}-15\right)
Variable c cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 10c, the least common multiple of 5,2c.
-6c=5c^{2}-75
Use the distributive property to multiply 5 by c^{2}-15.
-6c-5c^{2}=-75
Subtract 5c^{2} from both sides.
-6c-5c^{2}+75=0
Add 75 to both sides.
-5c^{2}-6c+75=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.
c=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-5\right)\times 75}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -6 for b, and 75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
c=\frac{-\left(-6\right)±\sqrt{36-4\left(-5\right)\times 75}}{2\left(-5\right)}
Square -6.
c=\frac{-\left(-6\right)±\sqrt{36+20\times 75}}{2\left(-5\right)}
Multiply -4 times -5.
c=\frac{-\left(-6\right)±\sqrt{36+1500}}{2\left(-5\right)}
Multiply 20 times 75.
c=\frac{-\left(-6\right)±\sqrt{1536}}{2\left(-5\right)}
Add 36 to 1500.
c=\frac{-\left(-6\right)±16\sqrt{6}}{2\left(-5\right)}
Take the square root of 1536.
c=\frac{6±16\sqrt{6}}{2\left(-5\right)}
The opposite of -6 is 6.
c=\frac{6±16\sqrt{6}}{-10}
Multiply 2 times -5.
c=\frac{16\sqrt{6}+6}{-10}
Now solve the equation c=\frac{6±16\sqrt{6}}{-10} when ± is plus. Add 6 to 16\sqrt{6}.
c=\frac{-8\sqrt{6}-3}{5}
Divide 6+16\sqrt{6} by -10.
c=\frac{6-16\sqrt{6}}{-10}
Now solve the equation c=\frac{6±16\sqrt{6}}{-10} when ± is minus. Subtract 16\sqrt{6} from 6.
c=\frac{8\sqrt{6}-3}{5}
Divide 6-16\sqrt{6} by -10.
c=\frac{-8\sqrt{6}-3}{5} c=\frac{8\sqrt{6}-3}{5}
The equation is now solved.
-6c=5\left(c^{2}-15\right)
Variable c cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 10c, the least common multiple of 5,2c.
-6c=5c^{2}-75
Use the distributive property to multiply 5 by c^{2}-15.
-6c-5c^{2}=-75
Subtract 5c^{2} from both sides.
-5c^{2}-6c=-75
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.
\frac{-5c^{2}-6c}{-5}=-\frac{75}{-5}
Divide both sides by -5.
c^{2}+\left(-\frac{6}{-5}\right)c=-\frac{75}{-5}
Dividing by -5 undoes the multiplication by -5.
c^{2}+\frac{6}{5}c=-\frac{75}{-5}
Divide -6 by -5.
c^{2}+\frac{6}{5}c=15
Divide -75 by -5.
c^{2}+\frac{6}{5}c+\left(\frac{3}{5}\right)^{2}=15+\left(\frac{3}{5}\right)^{2}
Divide \frac{6}{5}, the coefficient of the x term, by 2 to get \frac{3}{5}. Then add the square of \frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
c^{2}+\frac{6}{5}c+\frac{9}{25}=15+\frac{9}{25}
Square \frac{3}{5} by squaring both the numerator and the denominator of the fraction.
c^{2}+\frac{6}{5}c+\frac{9}{25}=\frac{384}{25}
Add 15 to \frac{9}{25}.
\left(c+\frac{3}{5}\right)^{2}=\frac{384}{25}
Factor c^{2}+\frac{6}{5}c+\frac{9}{25}. 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(c+\frac{3}{5}\right)^{2}}=\sqrt{\frac{384}{25}}
Take the square root of both sides of the equation.
c+\frac{3}{5}=\frac{8\sqrt{6}}{5} c+\frac{3}{5}=-\frac{8\sqrt{6}}{5}
Simplify.
c=\frac{8\sqrt{6}-3}{5} c=\frac{-8\sqrt{6}-3}{5}
Subtract \frac{3}{5} from both sides of the equation.
Examples
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4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
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Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
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