Quadratic Equations: Positive Non-Perfect Square Discriminant Results In Two Distinct Irreducible Roots

if the discriminant is positive, > 0 but NOT a perfect square

irrational

If the discriminant is positive but not a perfect square, then the roots of the quadratic equation will be two distinct irrational numbers. This means that the quadratic equation will have two roots, but these roots cannot be simplified as whole numbers or fractions.

Let’s consider the quadratic equation with the general form ax^2 + bx + c = 0. The discriminant of this equation is b^2 – 4ac. If the discriminant is positive and not a perfect square, then we have:

b^2 – 4ac > 0

b^2 > 4ac

We can then use the quadratic formula to find the roots of the equation:

x = [-b ± √(b^2 – 4ac)] / 2a

Since the discriminant is positive, the square root term will be a positive number. However, since the discriminant is not a perfect square, we cannot simplify this square root term.

For example, consider the quadratic equation x^2 + 3x + 2 = 0. The discriminant of this equation is:

b^2 – 4ac = 3^2 – 4(1)(2) = 1

Since the discriminant is positive but not a perfect square, we know that the roots of the equation will be two distinct irrational numbers. Using the quadratic formula, we find:

x = [-3 ± √1] / 2

x = (-3 + √1) / 2 or x = (-3 – √1) / 2

Simplifying, we get:

x = -1 or x = -2

So the roots of the equation are two distinct irrational numbers (-1 and -2), and are not whole numbers or fractions.

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