Type a math problem
Solve for x
To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation , where and are the solutions of the quadratic equation .
All equations of the form can be solved using the quadratic formula: . Substitute for , for , and for in the quadratic formula.
Do the calculations.
Solve the equation when is plus and when is minus.
Rewrite the inequality by using the obtained solutions.
For the product to be , one of the values and has to be and the other has to be . Consider the case when is and is .
This is false for any .
Consider the case when is and is .
The solution satisfying both inequalities is .
The final solution is the union of the obtained solutions.
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Graph Inequality
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x^{2}-7x+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(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 1\times 12}}{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, -7 for b, and 12 for c in the quadratic formula.
x=\frac{7±1}{2}
Do the calculations.
x=4 x=3
Solve the equation x=\frac{7±1}{2} when ± is plus and when ± is minus.
\left(x-4\right)\left(x-3\right)\leq 0
Rewrite the inequality by using the obtained solutions.
x-4\geq 0 x-3\leq 0
For the product to be ≤0, one of the values x-4 and x-3 has to be ≥0 and the other has to be ≤0. Consider the case when x-4 is ≥0 and x-3 is ≤0.
x\in \emptyset
This is false for any x.
x-3\geq 0 x-4\leq 0
Consider the case when x-3 is ≥0 and x-4 is ≤0.
x\in \begin{bmatrix}3,4\end{bmatrix}
The solution satisfying both inequalities is x\in \left[3,4\right].
x\in \begin{bmatrix}3,4\end{bmatrix}
The final solution is the union of the obtained solutions.