x 2 + 2 x - 15 &lt; 0

Question

x      2        +    2    x    -    15    &amp;lt;    0

Milton Hampton · Accepted Answer

Since             x      2        +    2    x    -    15    =          (      x      +      5      )              (      x      -      3      )      , we can suggest the following reasoning.The product of two real numbers can be negative if one of them is positive and another is negative.Therefore, we have two solutions:            A      1      ) x+5&amp;gt;0 AND x−3&amp;lt;0            A      2      ) x+5&amp;lt;0 AND x−3&amp;gt;0The case             A      1       defines x asx&amp;gt;−5 AND x&amp;lt;3, which defines an interval for x:−5

Taniya Melton · Accepted Answer

As we know, the graph of the quadratic polynomial on the left is parabola. Since the coefficient at

x^{2}

is positive, this parabola directs its endpoints upward.
Therefore, the only way it can be negative is in-between its roots, where it's equal to zero.
In other words, the solutions to a inequality is the area between solutions to equality

x^{2} + 2 x - 15 = 0

Obvious solutions are x=3 and x=−5.
So, the solutions are −5

x^2+2x−15<0

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