1Cademy - Using the Discriminant to Determine the Number of Solutions of Four Quadratic Equations

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Number of Solutions of a Quadratic Equation Based on the Discriminant

Example

Using the Discriminant to Determine the Number of Solutions of Four Quadratic Equations

Use the discriminant $b^2 - 4ac$ to determine how many real solutions each of the following quadratic equations has, without fully solving them.

ⓐ $2v^2 - 3v + 6 = 0$

The equation is already in standard form. Identify $a = 2$ , $b = -3$ , $c = 6$ . Compute the discriminant:

$(-3)^2 - 4 \cdot 2 \cdot 6 = 9 - 48 = -39$

Because the discriminant is negative, there are no real solutions.

ⓑ $3x^2 + 7x - 9 = 0$

Identify $a = 3$ , $b = 7$ , $c = -9$ . Compute the discriminant:

$(7)^2 - 4 \cdot 3 \cdot (-9) = 49 + 108 = 157$

Because the discriminant is positive, there are two real solutions.

Identify $a = 5$ , $b = 1$ , $c = 4$ . Compute the discriminant:

$(1)^2 - 4 \cdot 5 \cdot 4 = 1 - 80 = -79$

Because the discriminant is negative, there are no real solutions.

ⓓ $9y^2 - 6y + 1 = 0$

Identify $a = 9$ , $b = -6$ , $c = 1$ . Compute the discriminant:

$(-6)^2 - 4 \cdot 9 \cdot 1 = 36 - 36 = 0$

Because the discriminant is zero, there is one real solution.

This example illustrates all three possible outcomes of the discriminant in a single exercise. When $b^2 - 4ac$ is negative (parts ⓐ and ⓒ), the square root of a negative number is not real, so the Quadratic Formula produces no real solutions. When $b^2 - 4ac$ is positive (part ⓑ), the $\pm$ in the formula yields two distinct values. When $b^2 - 4ac$ equals zero (part ⓓ), the $\pm$ contributes nothing, and the formula produces exactly one solution.

Updated 2026-04-21

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OpenStax Elementary Algebra

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