Quadratic Formula, Discriminant & Vieta's

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✖️ 1. Deriving and applying the Quadratic Formula to find exact roots

🔑 The Quadratic Formula

For any quadratic $ax^2 + bx + c = 0$ where $a \neq 0$ , the roots are $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
The plus-minus symbol $\pm$ gives you two solutions in one formula.
Always identify $a$ , $b$ , and $c$ first before plugging into the formula.
The formula works even when factoring is impossible or messy.
If $a$ is negative, factor it out or keep careful track of signs.

Example: Solve $2x^2 - 5x + 2 = 0$ . Here $a=2$ , $b=-5$ , $c=2$ . Then $x = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm 3}{4}$ , so $x = 2$ or $x = 0.5$ .

💡 Memory hook: "Negative b, plus or minus the square root, all over 2a."

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1. Deriving and applying the Quadratic Formula to find exact roots

Deriving and Applying the Quadratic Formula

The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ provides the exact roots of any quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$ . It is derived by completing the square on the general form.

Intuition: The formula encodes all possible solutions by accounting for the parabola's vertex position and width, yielding two roots (possibly equal or complex) through the $\pm$ operation.

Core Rules:

Applicability: Valid only when $a \neq 0$ (otherwise the equation is linear).
Two solutions: The $\pm$ symbol generates both roots simultaneously.
Exact form: Roots may be irrational or complex; leave in radical form unless approximation is requested.
Sign of $a$ : Does not affect root existence, only parabola orientation.

Consequence: Every quadratic equation has exactly two roots in the complex number system, though they may coincide or be non-real.

Example: For $2x^2 - 4x - 6 = 0$ , we have $x = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm 8}{4}$ , giving $x = 3$ or $x = -1$ .

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A student is solving the equation $x^2 - 5x - 6 = 0$ using the quadratic formula. They write their first step as:

$x = (5 \pm \sqrt{25 - 24}) / 2$

Based on the core rules of the formula, what is their specific error?

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✖️ 2. Using the Discriminant to determine the number and nature of roots

🔍 The Discriminant Test

The discriminant is $D = b^2 - 4ac$ (the part under the square root).
If $D > 0$ , you get two distinct real roots.
If $D = 0$ , you get exactly one real root (a repeated root).
If $D < 0$ , you get no real roots (two complex roots).
You can check the discriminant before solving to know what to expect.

Example: For $x^2 + 4x + 5 = 0$ , compute $D = 16 - 20 = -4 < 0$ , so no real solutions exist.

💡 Visual cue: Positive D = parabola crosses x-axis twice; Zero D = touches once; Negative D = floats above or below.

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2. Using the Discriminant to determine the number and nature of roots

Using the Discriminant ( $D = b^2 - 4ac$ )

The discriminant $D = b^2 - 4ac$ is the expression under the square root in the quadratic formula. It determines the number and type of roots without computing them explicitly.

Intuition: The discriminant measures whether the parabola intersects the $x$ -axis (real roots), touches it (repeated root), or misses it entirely (complex roots).

Core Rules:

$D > 0$ : Two distinct real roots.
$D = 0$ : Exactly one repeated real root (the vertex touches the $x$ -axis).
$D < 0$ : Two complex conjugate roots (no real intersections).
Perfect square $D$ : If $D$ is a perfect square, roots are rational; otherwise irrational.

Consequence: The discriminant provides immediate qualitative information about solutions, essential for analyzing feasibility in applied problems.

Example: For $x^2 + 2x + 5 = 0$ , we have $D = 4 - 20 = -16 < 0$ , so the equation has two complex roots and no real solutions.

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Calculate the discriminant of the quadratic equation $3x^2 - 2x - 5 = 0$ .

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✖️ 3. Using Vieta's formulas to find sums and products of roots without solving

🧮 Vieta's Shortcut Formulas

For $ax^2 + bx + c = 0$ with roots $r$ and $s$ , sum of roots is $r + s = -\frac{b}{a}$ .
The product of roots is $rs = \frac{c}{a}$ .
These formulas let you find relationships without computing the actual roots.
Useful for checking answers or solving problems about root properties.
Convention: Write the equation in standard form first so you identify $a$ , $b$ , $c$ correctly.

Example: For $3x^2 - 12x + 9 = 0$ , sum $= \frac{12}{3} = 4$ and product $= \frac{9}{3} = 3$ .

💡 Memory hook: Sum uses $-b$ , product uses $c$ , both divided by $a$ .

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3. Using Vieta's formulas to find sums and products of roots without solving

Using Vieta's Formulas

Vieta's formulas relate the coefficients of $ax^2 + bx + c = 0$ to its roots $r_1$ and $r_2$ without solving: $r_1 + r_2 = -\frac{b}{a}$ and $r_1 \cdot r_2 = \frac{c}{a}$ . These follow from expanding $(x - r_1)(x - r_2) = 0$ .

Intuition: The sum of roots depends on the linear coefficient, while the product depends on the constant term, both scaled by the leading coefficient.

Core Rules:

Sum: $r_1 + r_2 = -\frac{b}{a}$ (note the negative sign).
Product: $r_1 \cdot r_2 = \frac{c}{a}$ .
Validity: Holds for all roots, real or complex.
Sign interpretation: If $\frac{c}{a} > 0$ , roots have the same sign; if $\frac{c}{a} < 0$ , opposite signs.

Consequence: Vieta's formulas enable rapid analysis of root properties and verification of solutions without explicit computation.

Example: For $3x^2 - 12x + 9 = 0$ , the sum is $-\frac{-12}{3} = 4$ and product is $\frac{9}{3} = 3$ , so roots are $r_1 = 3, r_2 = 1$ .

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For the equation $2x^2 - 10x + 7 = 0$ , what is the sum of its roots?

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✖️ 4. Reconstructing a quadratic equation given its roots

🔨 Building Quadratics from Roots

If roots are $r$ and $s$ , the quadratic is $(x - r)(x - s) = 0$ .
Expand to get $x^2 - (r+s)x + rs = 0$ using Vieta's formulas in reverse.
You can multiply the entire equation by any nonzero constant and it remains valid.
This method works for any two numbers, real or complex.
Always expand and simplify to standard form.

Example: Roots are 2 and -3. Then $(x - 2)(x + 3) = x^2 + x - 6 = 0$ .

💡 Quick check: Plug each root back into your equation to verify it equals zero.

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4. Reconstructing a quadratic equation given its roots

Reconstructing a Quadratic Equation from Roots

Given roots $r_1$ and $r_2$ , the quadratic equation is $(x - r_1)(x - r_2) = 0$ , which expands to $x^2 - (r_1 + r_2)x + r_1 r_2 = 0$ . This is the reverse application of Vieta's formulas.

Intuition: A quadratic is uniquely determined (up to scaling) by its two roots, as the factored form directly encodes the $x$ -intercepts.

Core Rules:

Factored form: Start with $(x - r_1)(x - r_2) = 0$ .
Expanded form: $x^2 - Sx + P = 0$ where $S = r_1 + r_2$ and $P = r_1 r_2$ .
Scaling: Multiply by any nonzero constant $a$ to obtain $ax^2 - aSx + aP = 0$ .
Irrational/complex roots: The method works identically; coefficients may be irrational or complex.

Consequence: This technique is essential for constructing equations from geometric or physical constraints.

Example: Given roots $5$ and $-2$ , we have $S = 3$ , $P = -10$ , yielding $x^2 - 3x - 10 = 0$ .

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A quadratic equation has roots $4$ and $7$ . Which of the following represents this equation in the expanded form $x^2 - Sx + P = 0$ ?

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✖️ 5. Applications: Analyzing stability and optimization with quadratics

🌍 Real-World Quadratic Applications

Population models: Quadratics model growth with limiting factors; roots show equilibrium points.
Pricing problems: Revenue $R = (price)(quantity)$ is often quadratic; maximum occurs at the vertex.
Use the discriminant to check if a model has real solutions (feasible scenarios).
Use Vieta's formulas to quickly find total or average outcomes without full solving.
The vertex formula $x = -\frac{b}{2a}$ finds optimal price or population level.

Example: Revenue $R = -2p^2 + 40p$ maximizes at $p = \frac{40}{4} = 10$ dollars per unit.

💡 Context cue: Negative $a$ means parabola opens down, so vertex is a maximum.

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5. Applications: Analyzing stability and optimization with quadratics

Applications: Stability and Optimization

Quadratic equations model equilibrium points in population dynamics (e.g., logistic growth) and revenue functions in economics. The discriminant determines feasibility, while Vieta's formulas reveal stability conditions.

Intuition: Real roots correspond to physically meaningful equilibria or break-even points; the discriminant indicates whether such states exist.

Core Rules:

Population models: Roots of $rN(1 - N/K) = h$ (harvest rate $h$ ) determine sustainable populations; $D < 0$ implies extinction.
Revenue optimization: For $R(p) = -ap^2 + bp$ , the vertex $p = \frac{b}{2a}$ maximizes revenue; roots show break-even prices.
Stability: In discrete models, if the product of roots (via Vieta) exceeds 1 in absolute value, equilibria are unstable.
Constraint analysis: $D = 0$ marks critical thresholds (e.g., maximum sustainable harvest).

Consequence: Quadratic analysis provides quantitative predictions for system behavior and optimal decision-making.

Example: For revenue $R(p) = -2p^2 + 40p$ , roots are $p = 0$ and $p = 20$ dollars; maximum revenue occurs at $p = 10$ dollars.

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A company's revenue function is given by $R(p) = -3p^2 + 60p$ , where $p$ is the price in dollars. Find the price $p$ that maximizes the revenue.

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Quadratic formula, Discriminant, and Vieta's formulas

✖️ 1. Deriving and applying the Quadratic Formula to find exact roots

🔑 The Quadratic Formula

1. Deriving and applying the Quadratic Formula to find exact roots

Deriving and Applying the Quadratic Formula

✖️ 2. Using the Discriminant to determine the number and nature of roots

🔍 The Discriminant Test

2. Using the Discriminant to determine the number and nature of roots

Using the Discriminant ( $D = b^2 - 4ac$ )

✖️ 3. Using Vieta's formulas to find sums and products of roots without solving

🧮 Vieta's Shortcut Formulas

3. Using Vieta's formulas to find sums and products of roots without solving

Using Vieta's Formulas

✖️ 4. Reconstructing a quadratic equation given its roots

🔨 Building Quadratics from Roots

4. Reconstructing a quadratic equation given its roots

Reconstructing a Quadratic Equation from Roots

✖️ 5. Applications: Analyzing stability and optimization with quadratics

🌍 Real-World Quadratic Applications

5. Applications: Analyzing stability and optimization with quadratics

Applications: Stability and Optimization

AWAITING_CONFIRMATION

✖️ 1. Deriving and applying the Quadratic Formula to find exact roots

🔑 The Quadratic Formula

1. Deriving and applying the Quadratic Formula to find exact roots

Deriving and Applying the Quadratic Formula

✖️ 2. Using the Discriminant to determine the number and nature of roots

🔍 The Discriminant Test

2. Using the Discriminant to determine the number and nature of roots

Using the Discriminant (D=b2−4acD = b^2 - 4acD=b2−4ac)

✖️ 3. Using Vieta's formulas to find sums and products of roots without solving

🧮 Vieta's Shortcut Formulas

3. Using Vieta's formulas to find sums and products of roots without solving

Using Vieta's Formulas

✖️ 4. Reconstructing a quadratic equation given its roots

🔨 Building Quadratics from Roots

4. Reconstructing a quadratic equation given its roots

Reconstructing a Quadratic Equation from Roots

✖️ 5. Applications: Analyzing stability and optimization with quadratics

🌍 Real-World Quadratic Applications

5. Applications: Analyzing stability and optimization with quadratics

Applications: Stability and Optimization

AWAITING_CONFIRMATION

Using the Discriminant ( $D = b^2 - 4ac$ )