Solve for x
x\in \left(4-2\sqrt{7},2\sqrt{7}+4\right)
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x^{2}-8x-12<0
Multiply the inequality by -1 to make the coefficient of the highest power in -x^{2}+8x+12 positive. Since -1 is negative, the inequality direction is changed.
x^{2}-8x-12=0
To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\left(-12\right)}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -8 for b, and -12 for c in the quadratic formula.
x=\frac{8±4\sqrt{7}}{2}
Do the calculations.
x=2\sqrt{7}+4 x=4-2\sqrt{7}
Solve the equation x=\frac{8±4\sqrt{7}}{2} when ± is plus and when ± is minus.
\left(x-\left(2\sqrt{7}+4\right)\right)\left(x-\left(4-2\sqrt{7}\right)\right)<0
Rewrite the inequality by using the obtained solutions.
x-\left(2\sqrt{7}+4\right)>0 x-\left(4-2\sqrt{7}\right)<0
For the product to be negative, x-\left(2\sqrt{7}+4\right) and x-\left(4-2\sqrt{7}\right) have to be of the opposite signs. Consider the case when x-\left(2\sqrt{7}+4\right) is positive and x-\left(4-2\sqrt{7}\right) is negative.
x\in \emptyset
This is false for any x.
x-\left(4-2\sqrt{7}\right)>0 x-\left(2\sqrt{7}+4\right)<0
Consider the case when x-\left(4-2\sqrt{7}\right) is positive and x-\left(2\sqrt{7}+4\right) is negative.
x\in \left(4-2\sqrt{7},2\sqrt{7}+4\right)
The solution satisfying both inequalities is x\in \left(4-2\sqrt{7},2\sqrt{7}+4\right).
x\in \left(4-2\sqrt{7},2\sqrt{7}+4\right)
The final solution is the union of the obtained solutions.
Examples
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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
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Limits
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