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The discriminant is a key part of the quadratic formula. It's the part under the square root symbol, given by: \[ \text{Discriminant} = b^2 - 4ac \] The discriminant helps us determine the nature of the roots of a quadratic equation. When we plug in the values of \(a\), \(b\), and \(c\) (the coefficients of the quadratic equation \(ax^2 + bx + c = 0\)), the discriminant tells us whether the roots are real, complex, or identical. If the discriminant is:

• Positive (>0): There are two distinct real roots.
• Zero (=0): There is exactly one real root.
• Negative (<0): There are two complex roots (no real solutions).

So, the discriminant is like a tool that helps us quickly understand the type of solutions our quadratic equation will have.

The roots of quadratic equations are the solutions to the equation \(ax^2 + bx + c = 0\). These are the values of \(x\) that make the equation true. We find these roots using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here's how it works:

• First, find the discriminant \(b^2 - 4ac\).
• Next, take the square root of the discriminant.
• Then, use the \(\pm\) symbol to calculate two possible values of \(x\).

If the discriminant is positive, the roots will be two different real numbers. If the discriminant is zero, there will be just one root (both calculations give the same result). If the discriminant is negative, the roots will be complex (involving imaginary numbers), meaning there are no real roots. Understanding this helps in quickly determining the type and number of solutions your quadratic equation will possess.

When the discriminant \(\Delta\) is zero (\(b^2 - 4ac = 0\)), the quadratic equation has exactly one solution. This is called a 'single solution' or a 'repeated root.' Here's why: Remember the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] When \(b^2 - 4ac = 0\): \[ x = \frac{-b \pm \sqrt{0}}{2a} \] Since \(\sqrt{0}\) is zero, this simplifies to: \[ x = \frac{-b}{2a} \] Notice there's no '±' anymore because adding or subtracting zero doesn't change the value. Therefore, you only get one unique root. This single solution means the parabola described by our quadratic equation touches the x-axis at just one point, indicating that the vertex of the parabola lies exactly on the x-axis. This scenario contrasts with having two different roots, where the parabola would intersect the x-axis at two distinct points, or complex roots, where it doesn't touch the x-axis at all.