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x^{2}+2x-3=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{-2±\sqrt{2^{2}-4\times 1\left(-3\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, 2 for b, and -3 for c in the quadratic formula.
x=\frac{-2±4}{2}
Do the calculations.
x=1 x=-3
Solve the equation x=\frac{-2±4}{2} when ± is plus and when ± is minus.
\left(x-1\right)\left(x+3\right)\leq 0
Rewrite the inequality by using the obtained solutions.
x-1\geq 0 x+3\leq 0
For the product to be ≤0, one of the values x-1 and x+3 has to be ≥0 and the other has to be ≤0. Consider the case when x-1\geq 0 and x+3\leq 0.
x\in \emptyset
This is false for any x.
x+3\geq 0 x-1\leq 0
Consider the case when x-1\leq 0 and x+3\geq 0.
x\in \begin{bmatrix}-3,1\end{bmatrix}
The solution satisfying both inequalities is x\in \left[-3,1\right].
x\in \begin{bmatrix}-3,1\end{bmatrix}
The final solution is the union of the obtained solutions.