Understanding Quadratic Equations: Explaining the Relationship Between b²-4ac=0 and Equal Roots

If b²-4ac = 0

If b²-4ac = 0, then it means that the quadratic equation of the form ax² + bx + c = 0 has equal roots

If b²-4ac = 0, then it means that the quadratic equation of the form ax² + bx + c = 0 has equal roots.

To understand why this is the case, let’s break down the quadratic formula and the discriminant.

The quadratic formula is given by:
x = (-b ± √(b²-4ac))/2a

The discriminant, which is the expression b²-4ac, is used to determine the nature of the roots. Here are the possible values for the discriminant:

1. If the discriminant is positive (b²-4ac > 0), then the quadratic equation has two distinct real roots.
2. If the discriminant is zero (b²-4ac = 0), then the quadratic equation has equal roots.
3. If the discriminant is negative (b²-4ac < 0), then the quadratic equation has no real roots, but instead has complex roots. In this case, since b²-4ac = 0, the equation has equal roots. This means that the quadratic equation has only one solution, which is repeated. Visualizing it geometrically, the graph of the equation would have its vertex touching the x-axis at a single point. In terms of solving the equation, the quadratic formula simplifies to: x = -b/2a By substituting the values into the formula, we can find the value of x. For example, if we have the quadratic equation x² - 4x + 4 = 0, we can see that b = -4, a = 1, and c = 4. Plugging these values into the formula, we get: x = -(-4)/(2*1) = 4/2 = 2 Therefore, the equation x² - 4x + 4 = 0 has the repeated root x = 2. Remember that when b²-4ac = 0, the quadratic equation has equal roots, which means that there is only one solution.

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