4.3 The Discriminant & Nature of Roots

4.3 The Discriminant & Nature of Roots

Quick Summary: In the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, the expression under the square root, $b^2 - 4ac$, is called the discriminant (denoted by $\Delta$). Because you cannot take the real square root of a negative number, the value and sign of $\Delta$ completely dictate the number and nature of real solutions without needing to solve the equation.

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The Three Discriminant Regimes

   CASE 1: Delta > 0             CASE 2: Delta = 0             CASE 3: Delta < 0
Two Distinct Real Roots         One Repeated Real Root         No Real Roots (Complex)
      y                             y                             y
      |     /\                      |     /\                      |       /\
      |    /  \                     |    /  \                     |      /  \
------+---*----*----x         ------+---*-------x         --------+-----/----\----x
      |  /      \                   |  / \                        |    /      \
      | /        \                  | /   \                       |   /        \
      |                             |                             |  * Vertex > 0
  2 x-intercepts                 1 x-intercept (tangent)        0 x-intercepts

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Complete Discriminant Classification Matrix

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Parameter Problems on the Digital SAT

The Digital SAT frequently tests the discriminant in reverse: you are given information about the number of solutions and asked to find an unknown constant $c$, $k$, or $b$.

SAT Trigger Phrases & Setup Rules

+----------------------------------------------+----------------------------------+
| SAT Problem Phrasing                         | Mathematical Setup               |
+----------------------------------------------+----------------------------------+
| "has exactly one real solution"              | Set b^2 - 4ac = 0                 |
| "is tangent to the x-axis"                   | Set b^2 - 4ac = 0                 |
| "has two distinct real solutions"           | Set b^2 - 4ac > 0                 |
| "intersects the x-axis twice"                | Set b^2 - 4ac > 0                 |
| "has no real solutions"                      | Set b^2 - 4ac < 0                 |
| "does not intersect the x-axis"              | Set b^2 - 4ac < 0                 |
| "has at least one real solution"             | Set b^2 - 4ac >= 0                |
+----------------------------------------------+----------------------------------+

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Worked Examples: Step-by-Step Parameter Calculations

Worked Example 1: Finding an Unknown Constant for Exactly One Solution

Problem: In the equation $3x^2 - 12x + k = 0$, $k$ is a constant. If the equation has exactly one real solution, what is the value of $k$?

• Step 1: Identify coefficients: $a = 3, b = -12, c = k$.
• Step 2: Set the discriminant equal to zero: $$\Delta = b^2 - 4ac = 0$$ $$(-12)^2 - 4(3)(k) = 0$$
• Step 3: Solve for $k$: $$144 - 12k = 0 \implies 12k = 144 \implies k = 12$$

Worked Example 2: Inequality for No Real Solutions

Problem: For what values of $b$ does the equation $2x^2 + bx + 8 = 0$ have no real solutions?

• Step 1: Identify coefficients: $a = 2, b = b, c = 8$.
• Step 2: Apply the condition for no real solutions ($\Delta < 0$): $$b^2 - 4(2)(8) < 0$$ $$b^2 - 64 < 0$$
• Step 3: Solve the quadratic inequality: $$b^2 < 64 \implies -8 < b < 8$$

Worked Example 3: Discriminant in Non-Standard Form

Problem: The equation $x^2 - 6x + 13 = c$ has no real solutions. What is the range of possible values for the constant $c$?

• Step 1: Convert to standard form $ax^2 + bx + (13 - c) = 0$. Here $a = 1, b = -6, c_{\text{term}} = 13 - c$.
• Step 2: Apply $\Delta < 0$: $$(-6)^2 - 4(1)(13 - c) < 0$$ $$36 - 52 + 4c < 0$$ $$-16 + 4c < 0 \implies 4c < 16 \implies c < 4$$

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Discriminant Applied to Linear-Quadratic Systems

When a line $y = mx + d$ intersects a parabola $y = ax^2 + bx + c$, set them equal:

$$ax^2 + bx + c = mx + d \implies ax^2 + (b - m)x + (c - d) = 0$$

Compute the discriminant $\Delta = (b - m)^2 - 4a(c - d)$:

• $\Delta > 0$: The line intersects the parabola at two distinct points (secant line).
• $\Delta = 0$: The line is tangent to the parabola at exactly one point.
• $\Delta < 0$: The line never intersects the parabola (zero solutions).

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Desmos Discriminant Visualization Strategy

1. Slider Verification: If a problem contains an unknown constant $k$, type y = 3x^2 - 12x + k into Desmos.
2. Add Slider: Desmos will prompt to add a slider for k.
3. Drag the Slider: Move $k$ and observe how the parabola shifts vertically:
   • When $k < 12$: Parabola crosses $x$-axis twice ($2$ real solutions).
   • When $k = 12$: Parabola touches the $x$-axis at its vertex $(2, 0)$ ($1$ real solution).
   • When $k > 12$: Parabola floats entirely above the $x$-axis ($0$ real solutions).

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Common Pitfalls & Exam Traps

• Sign errors with negative $b$: When calculating $(-8)^2$, remember that $(-8)^2 = +64$, never $-64$.
• Forgetting to subtract non-zero right side: In $2x^2 + 4x = -c$, you must write $2x^2 + 4x + c = 0$ before identifying $a, b, c$.
• Confusing "no real solutions" with "no solutions": Quadratics with $\Delta < 0$ still have two complex solutions; they simply have zero real solutions.
• Forgetting the strict inequality: "Two distinct real solutions" requires $\Delta > 0$ (strictly greater), not $\Delta \ge 0$.