What is the discriminant of 2x^2 + 3x - 5 = 0, and what does it tell us about the solutions?

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Accepted Answer

Discriminant is 49; two distinct real solutions. The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. For 2x^2 + 3x - 5 = 0, we have a = 2, b = 3, and c = -5. Substituting: D = 3^2 - 4(2)(-5) = 9 - (-40) = 9 + 40 = 49.

Since the discriminant is positive (49 > 0), the equation has two distinct real solutions. A common error is computing -4(2)(-5) as -40 instead of +40, which would give D = 9 - 40 = -31 ("Discriminant is -31; two complex solutions"), incorrectly suggesting complex solutions. Note that 49 is also a perfect square (7^2), which means the solutions are rational numbers  -  the equation can be factored over the rationals.

The discriminant is a powerful tool tested on the CLEP exam. If D > 0, there are two distinct real solutions. If D = 0, there is exactly one repeated real solution. If D < 0, there are two complex conjugate solutions (no real solutions). The discriminant also appears under the radical in the quadratic formula: x = (-b +/- sqrt(D)) / (2a). A perfect square discriminant means the solutions are rational and the quadratic can be factored with integer coefficients.

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