Determine the number of real solutions to a quadratic (e.g., via discriminant or graph). | Nonlinear equations in one variable | SAT Math

The discriminant tells you how many real roots a quadratic has without solving it.

核心知识

The discriminant $𝑏 2 - 4 𝑎 𝑐$ tells you how many real solutions $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 = 0$ has: positive means two solutions, zero means one solution, negative means no real solutions. You don't need to solve — just compute one number.

深入理解

The discriminant is a quick count check for a quadratic.

Discriminant rule: $𝐷 = 𝑏 2 - 4 𝑎 𝑐$ .

$𝐷 > 0$ : two distinct real solutions
$𝐷 = 0$ : one real solution
$𝐷 < 0$ : no real solutions

If a parameter is in the equation, set $𝐷$ to the condition the question asks for and solve for that parameter. On a graph, the same count shows up as the number of x-intercepts.

分步讲解

Write the equation in standard form $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 = 0$ .
Identify $𝑎$ , $𝑏$ , and $𝑐$ .
Compute $𝐷 = 𝑏 2 - 4 𝑎 𝑐$ .
Compare $𝐷$ to 0: positive means 2 solutions, zero means 1, negative means 0.

常见误解

Confusing the discriminant with the full quadratic formula. You only need $𝑏 2 - 4 𝑎 𝑐$ , not the whole fraction.
Forgetting to rearrange the equation to standard form before reading off $𝑎$ , $𝑏$ , and $𝑐$ .
Mixing up the conditions: $𝐷 = 0$ means one solution (not zero solutions).

题目

示例解析

For what value of $𝑘$ does $𝑥 2 + 𝑘 𝑥 + 9 = 0$ have exactly one real solution, where $𝑘 > 0$ ?

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