Problem

Analyze \(x^{2}-6x+9=0\) through its discriminant. Identify \(a\), \(b\), and \(c\) with signs, compute \(D\), classify the number and type of roots, and state the corresponding \(x\text{-}\text{axis}\) behavior.

Big Picture

What this problem is really about

The discriminant classifies the root structure and its graph behavior before we need a full solution. We’ll identify the signed coefficients, evaluate b squared minus four a c, then use the result’s sign and square status to determine how many real roots there are, what type they are, and how the parabola meets the x-axis.

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a = 1, b = −6, c = 9; D = 36 − 36 = 0; two distinct real roots; the parabola has two x-intercepts.

a = 1, b = −6, c = 9; D = 36 − 36 = 0; one repeated real rational root; the parabola is tangent to the x-axis.

a = 1, b = −6, c = 9; D = 0; two distinct rational roots; the parabola has two x-intercepts.

a = 1, b = −6, c = 9; D = −36; no real roots; the parabola has no x-intercepts.

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Analyze \(x^{2}-6x+9=0\) through its discriminant. Identify \(a\), \(b\), and \(c\) with signs, compute \(D\), classify the number and type of roots, and state the corresponding \(x\text{-}\text{axis}\) behavior.

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