Final answer:
The discriminant of a quadratic equation, represented by b²-4ac, dictates the number of real roots the equation has. A positive discriminant indicates two real roots, zero discriminant results in one real root, and a negative discriminant means there are no real roots. An example with x² + 2x - 3 = 0 confirms this relationship, revealing two real roots and corresponding to the case of a parabola crossing the x-axis twice.
Step-by-step explanation:
Relationship Between Discriminant and Real Roots of Quadratic Equations
The relationship between the discriminant of a quadratic equation and the number of real roots it possesses is fundamental in determining the nature of its solutions. A quadratic equation is generally expressed in the form ax²+bx+c = 0, where 'a', 'b', and 'c' are constants and 'a' is not zero. We can determine the nature of the roots using the discriminant, which is the part of the quadratic formula under the square root and is given by b²-4ac.
For Case 1, where the parabola crosses the x-axis twice, the discriminant is positive (b²-4ac > 0), indicating two distinct real roots. For Case 2, where the parabola touches the x-axis exactly once, the discriminant is zero (b²-4ac = 0), resulting in exactly one real root. For Case 3, if the parabola does not cross the x-axis, the discriminant is negative (b²-4ac < 0), and there are no real roots, only complex ones.
Let's consider a specific example:
x² + 2x - 3 = 0
Here, a=1, b=2, and c=-3. Computing the discriminant gives us 2² - 4(1)(-3) = 4 + 12 = 16, which is positive. Thus, applying the quadratic formula, we will get two real solutions which we find to be x = 1 and x = -3. This corresponds to Case 1, with the parabola crossing the x-axis twice.
Physical relevance of the roots must also be considered in practical applications. Quadratic equations constructed on physical data often have real roots; however, only the positive values may be significant, depending on the context, such as when dealing with time or distances.