4.4. Nature of Roots

The equation ax² + bx + c = 0 are given by x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This chunk introduces the quadratic formula that is used to find the roots of a quadratic equation of the form ax² + bx + c = 0. The formula shows how to calculate the roots ( 'values' of x) based on the coefficients a, b, and c from the quadratic equation. The '±' symbol indicates that there can be two possible results for x, which correspond to the two potential solutions (roots) of the equation.

Think of the quadratic equation like a path that branches off into two different routes. The quadratic formula tells you how to find the point at which you can make a choice between these two paths, represented by the two solutions for x, which can lead to different outcomes.

If b² - 4ac > 0, we get two distinct real roots. If b² - 4ac = 0, then the roots are equal (two equal real roots). If b² - 4ac < 0, then there are no real roots.

In this chunk, we discuss the discriminant (b² - 4ac), which is a key value that indicates the nature of the roots of the quadratic equation. Specifically: 1. If the discriminant is greater than 0, this means the quadratic has two different real roots. 2. If the discriminant is exactly 0, the quadratic has one real root that is counted twice (commonly called repeated roots). 3. If the discriminant is less than 0, there are no real roots—this means that the solutions involve complex numbers.

Imagine you're throwing a ball in the air. The different outcomes of where the ball will land (i.e., will it bounce twice? will it land in the ground without bouncing? or will it lift off into a different dimension?) can be compared to the types of roots depending on the discriminant's value.

Example 7: Find the discriminant of the quadratic equation 2x² - 4x + 3 = 0, and hence find the nature of its roots. Solution: The given equation is of the form ax² + bx + c, where a = 2, b = -4, c = 3. Therefore, the discriminant b² - 4ac = (-4)² - (4 × 2 × 3) = 16 - 24 = -8 < 0. So, the given equation has no real roots.

This example illustrates how to compute the discriminant for a specific quadratic equation to determine its nature. By substituting the coefficients from the quadratic equation into the discriminant formula, we can find that the result is -8, which implies that the quadratic equation has no real roots. This step is fundamental in understanding how to analyze the nature of different quadratic equations.

Think of trying to find a treasure map. If you find the map leads to a spot underwater (like having no real roots), it wouldn't be possible to reach the treasure (real solutions) with normal means—indicating the treasure isn’t there in the real world.

Example 8: A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. ... So, the distance ‘x’ of the pole from gate B satisfies the equation x² + 7x - 60 = 0...

In this example, we analyze a practical situation involving placements in geometry, using the quadratic equation derived from the conditions given (distances to gates). After setting up the equation based on the scenario, we find the discriminant to determine the feasibility of placing the pole according to the conditions. It concludes whether the arrangement at that specific point is viable based on the quadratic roots.

Imagine you are setting up tents along a line (the boundary of the park) between two markers (the gates). Depending on how far apart the markers are and the rules you've set (like the 7-meter distance), you need to calculate the exact spots to set up your tents to ensure they fit without breaking the rules.

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