What is the discriminant of a quadratic function? What does it tell you about the graph of the function? Be specific.

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What is the discriminant of a quadratic function? What does it tell you about the graph of the function? Be specific.

Mathematics
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the discriminant for a quadratic \[ax^2 + bx + c = 0\] \[\Delta = b^2 - 4ac\] the discriminant tells you about the number of types of zeros of a quadratic equation there are 3 cases 1. \[\Delta > 0\] the quadratic has 2 unequal real roots... if the discriminant is a square number the roots are real and rational, otherwise there are irrational. 2. \[\Delta = 0\] the roots are real and repeated. This is the case of a perfect square (x +3)^2 = 0 -3 is a repeated root. 3. \[\Delta < 0\] the quadratic has no real roots. The roots are complex numbers... so the parabola is ways positive or always negative... hope it helps
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The discriminant isn't really that weird of a thing. If you want to solve for when x=0 of this equation: \[ax^2+bx+c=0\] Then we know that we can complete the square on it or memorize this equation for the roots: \[x= \frac{-b \pm \sqrt{b^2-4ac}}{2a}\] So we can see that when \(b^2-4ac\) is a positive number we'll have two roots, \(x= \frac{-b +\sqrt{b^2-4ac}}{2a}\) and \(x= \frac{-b - \sqrt{b^2-4ac}}{2a}\) just like we expect. If \(b^2-4ac\) is equal to zero, then the plus and minus part doesn't give us different answers since +0 and -0 are just zero, see: \(x= \frac{-b \pm 0}{2a} \) So now the only other case is when \(b^2-4ac\) is less than 0, you'll take the square root of a negative number! So depending on if you're familiar with complex numbers or not then that means your answer will have imaginary numbers in it or maybe your teacher just says that no solutions exist since you're only looking for real solutions.

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