Quadratic Equations: The Quadratic Formula and the Discriminant

Quadratic Equations: The Quadratic Formula and the Discriminant

Introduction

Quadratic equations are polynomial equations of degree two, typically written in the form ax2 + bx + c = 0, where a ≠ 0. This section explores methods for solving quadratic equations using the Quadratic Formula and analyzes the nature of their solutions using the discriminant.

Using the Quadratic Formula

Definition and Application

The Quadratic Formula provides a universal method for solving any quadratic equation of the form ax2 + bx + c = 0. The solutions are given by:

• Quadratic Formula:

$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$

• a, b, c are real numbers, with a ≠ 0.

• The symbol "±" indicates two possible solutions: one with addition, one with subtraction.

Example 1: Real Roots, c Positive

Problem: Solve 0 = -4.9t2 + 11.7t + 42 for t.

• Identify coefficients: a = -4.9, b = 11.7, c = 42

• Apply the formula:

$t = \frac{ -11.7 \pm \sqrt{ (11.7)^2 - 4(-4.9)(42) } }{ 2(-4.9) }$

$= \frac{ -11.7 \pm \sqrt{ 136.89 + 823.2 } }{ -9.8 }$

$= \frac{ -11.7 \pm \sqrt{ 960.09 } }{ -9.8 }$

$= \frac{ 11.7 \pm \sqrt{ 960.09 } }{ 9.8 }$

• Approximate solutions: t ≈ 4.4 seconds and t ≈ -2.0 seconds.

• Interpretation: Only the positive root is meaningful in this context (time after launch).

Example 2: Real Roots, c Negative

Problem: Solve x2 + 4x - 17 = 0.

• a = 1, b = 4, c = -17

$x = \frac{ -4 \pm \sqrt{ 4^2 - 4(1)(-17) } }{ 2(1) }$

$= \frac{ -4 \pm \sqrt{ 16 + 68 } }{ 2 }$

$= \frac{ -4 \pm \sqrt{ 84 } }{ 2 }$

$= \frac{ -4 \pm 2\sqrt{21} }{ 2 }$

$= -2 \pm \sqrt{21}$

• Two real, irrational roots.

Example 3: Complex Roots

Problem: Solve 5x2 + 8x + 11 = 0.

• a = 5, b = 8, c = 11

$x = \frac{ -8 \pm \sqrt{ 8^2 - 4(5)(11) } }{ 2(5) }$

$= \frac{ -8 \pm \sqrt{ 64 - 220 } }{ 10 }$

$= \frac{ -8 \pm \sqrt{ -156 } }{ 10 }$

$= \frac{ -8 \pm 2i\sqrt{39} }{ 10 }$

$= -0.8 \pm 0.2i\sqrt{39}$

• Two complex (non-real) roots.

Steps for Using the Quadratic Formula

1. Write the equation in standard form: ax2 + bx + c = 0.

2. Identify coefficients a, b, and c.

3. Substitute values into the Quadratic Formula.

4. Simplify under the square root (the discriminant).

5. Calculate both possible values for x.

The Discriminant

Definition and Role

The discriminant of a quadratic equation is the expression under the square root in the Quadratic Formula: b2 - 4ac. The value of the discriminant determines the number and type of roots of the equation.

Key Concept Table: Discriminant and Roots

Value of Discriminant	Type and Number of Roots	Description
$b^2 - 4ac > 0$ (perfect square)	2 real, rational roots	Roots are rational numbers; the graph crosses the x-axis at two points.
$b^2 - 4ac > 0$ (not a perfect square)	2 real, irrational roots	Roots are irrational numbers; the graph crosses the x-axis at two points.
$b^2 - 4ac = 0$	1 real, rational root	There is one repeated root; the graph touches the x-axis at one point (vertex).
$b^2 - 4ac < 0$	2 complex roots	No real roots; the graph does not cross the x-axis.

Example 4: Discriminant, Real Roots

Problem: Examine 2x2 - 10x + 7 = 0.

• a = 2, b = -10, c = 7

• Discriminant: $(-10)^2 - 4(2)(7) = 100 - 56 = 44$

• Since 44 > 0 and not a perfect square, there are two real, irrational roots.

Example 5: Discriminant, Complex Roots

Problem: Examine -5x2 + 10x - 15 = 0.

• a = -5, b = 10, c = -15

• Discriminant: $(10)^2 - 4(-5)(-15) = 100 - 300 = -200$

• Since the discriminant is negative, there are two complex roots.

Summary Table: Discriminant and Root Types

Discriminant Value	Number of Roots	Type of Roots
> 0 (perfect square)	2	Real, rational
> 0 (not perfect square)	2	Real, irrational
= 0	1	Real, rational (repeated root)
< 0	2	Complex (non-real)

Applications and Word Problems

Modeling with Quadratic Equations

• Quadratic equations are used to model real-world scenarios such as projectile motion, stopping distances, and geometry problems.

• For example, the height of a thrown object can be modeled by h(t) = -16t2 + vt + h0, where v is initial velocity and h0 is initial height.

Example: Projectile Motion

Problem: A football is thrown and its height is modeled by h = -16t2 + 45t + 4. To find when it hits the ground, set h = 0 and solve for t using the Quadratic Formula.

Example: Stopping Distance

Formula: $d = vt + \frac{ v^2 }{ 2 \mu g }$

• v: initial velocity (ft/s)

• t: reaction time (s)

• μ: coefficient of friction

• g: acceleration due to gravity (32 ft/s2)

Tables can be used to compare stopping distances for different velocities and conditions.

Practice Problems

Solving Quadratic Equations

• Use the Quadratic Formula to solve equations such as:

• $x^2 + 8x + 15 = 0$

• $12x^2 - 22x + 6 = 0$

• $4x^2 - 6x = -2$ (first rewrite as $4x^2 - 6x + 2 = 0$)

Analyzing the Discriminant

• Find the discriminant for equations like $x^2 - 8x + 16 = 0$ and describe the number and type of roots.

Common Questions and Higher-Order Thinking

• How does the discriminant determine the type of roots?

• Is it possible for a quadratic equation to have zero real or complex roots? (No; every quadratic has two roots, which may be real or complex.)

• Can the sign of coefficients a and c predict the nature of the roots? (If a and c have different signs, the product ac is negative, which often leads to a positive discriminant and real roots.)

Summary

• The Quadratic Formula is a universal method for solving quadratic equations.

• The discriminant $b^2 - 4ac$ determines the number and type of roots.

• Quadratic equations model many real-world phenomena, and understanding their solutions is essential in algebra.

Additional info: Some context and explanations have been expanded for clarity and completeness, including the summary tables and step-by-step examples.