2.1 Solving Quadratic Equations
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Learning Objectives
- Recognize the standard form of a quadratic equation
- Solve quadratic equations by factorization
- Solve quadratic equations using the quadratic formula
- Solve quadratic equations by completing the square
- Determine the nature of roots using the discriminant
Key Concepts
Introduction to Quadratic Equations
A quadratic equation is a polynomial equation of the second degree, which can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
Quadratic equations appear frequently in mathematics and have applications in physics, engineering, economics, and many other fields.
There are several methods to solve quadratic equations, including factorization, using the quadratic formula, and completing the square.
Solving by Factorization
Factorization involves rewriting the quadratic expression as a product of two linear factors.
If we can factorize ax² + bx + c into (px + q)(rx + s), then the solutions to the equation ax² + bx + c = 0 are x = -q/p and x = -s/r.
This method works well when the quadratic expression can be easily factorized.
Example 1
Solve x² - 5x + 6 = 0 by factorization.
Solution:
We need to find two numbers that multiply to give 6 and add to give -5. These numbers are -2 and -3. So, x² - 5x + 6 = (x - 2)(x - 3) = 0 Therefore, x - 2 = 0 or x - 3 = 0 So, x = 2 or x = 3
Example 2
Solve 2x² + 7x - 4 = 0 by factorization.
Solution:
We need to find two numbers that multiply to give -8 (= 2 × -4) and add to give 7. These numbers are 8 and -1. So, 2x² + 7x - 4 = (2x - 1)(x + 4) = 0 Therefore, 2x - 1 = 0 or x + 4 = 0 So, x = 1/2 or x = -4
The Quadratic Formula
The quadratic formula provides a direct way to find the solutions to any quadratic equation.
For the equation ax² + bx + c = 0, the solutions are given by x = (-b ± √(b² - 4ac)) / (2a).
The term b² - 4ac is called the discriminant and determines the nature of the roots.
Example 1
Solve x² - 6x + 8 = 0 using the quadratic formula.
Solution:
Here, a = 1, b = -6, and c = 8. Using the formula x = (-b ± √(b² - 4ac)) / (2a): x = (6 ± √(36 - 32)) / 2 x = (6 ± √4) / 2 x = (6 ± 2) / 2 So, x = 4 or x = 2
Example 2
Solve 3x² - 2x - 1 = 0 using the quadratic formula.
Solution:
Here, a = 3, b = -2, and c = -1. Using the formula x = (-b ± √(b² - 4ac)) / (2a): x = (2 ± √(4 + 12)) / 6 x = (2 ± √16) / 6 x = (2 ± 4) / 6 So, x = 1 or x = -1/3
Completing the Square
Completing the square is a method that involves rewriting the quadratic expression in the form a(x + p)² + q.
This method is useful not only for solving quadratic equations but also for finding the vertex of a parabola and for other algebraic manipulations.
To complete the square for x² + bx + c, we rewrite it as (x + b/2)² + (c - b²/4).
Example 1
Solve x² - 6x + 5 = 0 by completing the square.
Solution:
Step 1: Rearrange to get x² - 6x = -5 Step 2: Take half the coefficient of x and square it: (-6/2)² = 9 Step 3: Add and subtract this value: x² - 6x + 9 - 9 = -5 Step 4: Rewrite as a perfect square: (x - 3)² - 9 = -5 Step 5: Solve for x: (x - 3)² = 4 Step 6: Take the square root: x - 3 = ±2 So, x = 5 or x = 1
Example 2
Solve 2x² - 12x + 10 = 0 by completing the square.
Solution:
Step 1: Divide by the coefficient of x²: x² - 6x + 5 = 0 Step 2: Rearrange: x² - 6x = -5 Step 3: Take half the coefficient of x and square it: (-6/2)² = 9 Step 4: Add and subtract this value: x² - 6x + 9 - 9 = -5 Step 5: Rewrite as a perfect square: (x - 3)² - 9 = -5 Step 6: Solve for x: (x - 3)² = 4 Step 7: Take the square root: x - 3 = ±2 So, x = 5 or x = 1
The Discriminant and Nature of Roots
The discriminant, b² - 4ac, determines the nature of the roots of a quadratic equation.
If the discriminant is positive, the equation has two distinct real roots.
If the discriminant is zero, the equation has one repeated real root (a double root).
If the discriminant is negative, the equation has two complex conjugate roots (no real roots).
Example 1
Determine the nature of the roots of x² - 4x + 4 = 0.
Solution:
Here, a = 1, b = -4, and c = 4. The discriminant is b² - 4ac = (-4)² - 4(1)(4) = 16 - 16 = 0. Since the discriminant is zero, the equation has one repeated real root. Using the quadratic formula: x = 4/2 = 2 So, the equation has a double root at x = 2.
Example 2
Determine the nature of the roots of 2x² + x + 1 = 0.
Solution:
Here, a = 2, b = 1, and c = 1. The discriminant is b² - 4ac = 1 - 4(2)(1) = 1 - 8 = -7. Since the discriminant is negative, the equation has two complex conjugate roots (no real roots).
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