91Ó°ÊÓ

The discriminant is a key tool in understanding the nature of the roots of a quadratic expression. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant is calculated as \(b^2 - 4ac\). The value of the discriminant tells us whether the roots of the quadratic equation are real or complex.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, which is repeated.
• If the discriminant is negative, the equation has two complex roots.

In our given quadratic expression, \(2x^2 + 2x + 3\), the discriminant was calculated to be \(-20\). This tells us that the quadratic has complex roots, since \(-20\) is negative.

In algebra, factorization refers to expressing a quadratic expression as a product of its linear factors. Whether or not a quadratic expression can be factored into real linear factors depends on its roots. If the discriminant is zero or positive, the quadratic is factorable over the real numbers. This is because you can find real roots to rewrite the quadratic as the product of two binomials.
However, if the discriminant is negative, as with the expression \(2x^2 + 2x + 3\) (where the discriminant is \(-20\)), the expression cannot be factored into real linear factors. Instead, it will have complex factors, which involve imaginary numbers.

Real roots are the x-values where the quadratic equation intersects the x-axis. For quadratic equations, these are the solutions to \(ax^2 + bx + c = 0\). The nature of these roots depends on the discriminant:

• When the discriminant is positive, you get two distinct real roots. The graph of the quadratic will intersect the x-axis at two points.
• When the discriminant is zero, you have exactly one real root (a repeated root). The graph of the quadratic touches the x-axis at a single point.

Since the expression \(2x^2 + 2x + 3\) has a negative discriminant, it does not have any real roots. Instead, it is said to have complex roots.

Complex roots arise in quadratic equations when the discriminant is negative. These roots are not real numbers and do not correspond to points where the graph crosses or touches the x-axis. Instead, they involve imaginary numbers and exist as a conjugate pair (for example, if one root is \(a + bi\), the other will be \(a - bi\)).
In our example with the quadratic expression, \(2x^2 + 2x + 3\), the discriminant is \(-20\), indicating complex roots. This means that the roots cannot be represented on the real number line, but rather involve imaginary units denoted by \(i\), where \(i = \sqrt{-1}\). These roots arise when solving the equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), revealing the deep and interesting nature of complex numbers in mathematics.