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Interactive Audio Lesson
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Factoring a Quadratic Equation
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Today, we're going to solve the quadratic equation \( x^2 + 5x + 6 = 0 \) by factoring. Can anyone remind me of what factors are?
Factors are numbers we can multiply to get another number.
Exactly! Here, we need two numbers that multiply to 6 and add to 5. Can you think of any?
How about 2 and 3?
Great! So we can write the equation as \( (x + 2)(x + 3) = 0 \). What do we do next?
Set each factor to zero? So \( x + 2 = 0 \) and \( x + 3 = 0 \).
Exactly! Solving these gives us the roots. What are they?
So \( x = -2 \) and \( x = -3 \)!
Excellent, everyone! Remember: when factoring, look for pairs of numbers that work. Now let's summarize: we factored the quadratic into binomials and solved for the roots!
Completing the Square
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Now let’s try the equation \( x^2 + 4x + 1 = 0 \) and solve it by completing the square. Who can recall what that means?
It’s about rearranging the equation to form a perfect square.
Correct! First, we move the constant to the other side: \( x^2 + 4x = -1 \). Now, to complete the square, we need to calculate half of 4 squared. What’s that?
That would be 4! So we add 4 to both sides.
Exactly! Now, we have \( (x + 2)^2 = 3 \). What’s our next move?
We take the square root of both sides to get \( x + 2 = \, \pm \sqrt{3} \)!
Fantastic! Now let’s isolate \( x \) to find the solutions.
So \( x = -2 \pm \sqrt{3} \)!
Well done! Summarizing: we completed the square to find the roots.
Using the Quadratic Formula
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Let’s tackle the equation \( 2x^2 - 4x - 6 = 0 \) using the quadratic formula. Recall the formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Who can tell me the values of a, b, and c here?
Here, \( a = 2 \), \( b = -4 \), and \( c = -6 \).
Exactly! Now let’s substitute these values into our formula step-by-step. What does that look like?
So, we have \( x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)} \).
Wonderful! Now simplify the discriminant first, what do you get?
That gives \( x = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm 8}{4} \).
Correct! So identifying the solutions, what do you find?
Then, we have \( x = 3 \) and \( x = -1 \)!
Exactly right! So to summarize: we employed the quadratic formula and identified our roots.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we provide a series of worked examples demonstrating different techniques for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. Each example illustrates step-by-step solutions to reinforce understanding of these methods.
Detailed
Worked Examples
In this section, we delve into practical applications of the methods for solving quadratic equations through systematic worked examples. Each example showcases a different approach:
- Factoring: The first example demonstrates solving a quadratic equation by factoring it into the product of two binomials. This method is beneficial when the quadratic can be easily decomposed, leading to quick solutions.
- Example: Solving \( x^2 + 5x + 6 = 0 \) results in \( (x + 2)(x + 3) = 0 \) leading to roots \( x = -2 \) and \( x = -3 \).
- Completing the Square: The second example illustrates the completing the square method, which is especially useful for deriving the vertex form of a quadratic function.
- Example: For the equation \( x^2 + 4x + 1 = 0 \), transforming it leads to \( (x + 2)^2 - 3 = 0 \), from which we can derive the solutions \( x = -2 \pm \sqrt{3} \).
- Quadratic Formula: The third example employs the quadratic formula, an effective method when dealing with complex coefficients or unable to factorize easily.
- Example: Solving the equation \( 2x^2 - 4x - 6 = 0 \) using the quadratic formula yields the solutions \( x = 3 \) and \( x = -1 \).
These examples highlight the versatility of quadratic solving techniques, emphasizing their importance in various mathematical and real-world applications.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Example 1: Factoring
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Solve:
𝑥² + 5𝑥 + 6 = 0
Solution:
(𝑥 + 2)(𝑥 + 3) = 0 ⇒ 𝑥 = −2, 𝑥 = −3
Detailed Explanation
In this example, we are given the quadratic equation x² + 5x + 6 = 0. To solve this, we will factor the quadratic expression. The goal is to find two binomials that multiply to form the original quadratic equation.
- Look for two numbers that multiply to give 6 (the constant term) and add up to 5 (the coefficient of x). These numbers are 2 and 3.
- Hence, we can write the equation as (x + 2)(x + 3) = 0.
- To find the values of x, we set each factor equal to zero:
- x + 2 = 0 ⇒ x = -2
- x + 3 = 0 ⇒ x = -3
Therefore, the solutions are x = -2 and x = -3.
Examples & Analogies
Think of this as finding the correct combination for a locked box. Just like how you need the right numbers to unlock the box (which represent the factors), here we found the right numbers (2 and 3) to unlock the solutions to our quadratic expression.
Example 2: Completing the Square
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Solve:
𝑥² + 4𝑥 + 1 = 0
Solution:
𝑥² + 4𝑥 + 4 - 4 + 1 = (𝑥 + 2)² - 3 = 0 ⇒ (𝑥 + 2)² = 3 ⇒ 𝑥 = −2 ± √3
Detailed Explanation
In this example, we solve the quadratic equation x² + 4x + 1 = 0 using the method of completing the square.
- We start by rearranging the equation to focus on the x² and x terms.
- The first step is to add and subtract the square of half of the coefficient of x (which is 4/2 = 2) inside the equation: x² + 4x + 4 - 4 + 1 = 0.
- This gives us the form: (x + 2)² - 3 = 0.
- Next, we isolate the squared term: (x + 2)² = 3.
- To solve for x, we take the square root of both sides: x + 2 = ±√3.
- Finally, we subtract 2 to find the two solutions: x = -2 + √3 and x = -2 - √3.
Examples & Analogies
Imagine you are trying to form a perfect square using blocks. You have some blocks already and want to organize them into a square formation. Completing the square allows you to see how many additional blocks you need to form that perfect square while aligning everything neatly, just like manipulating the equation here.
Example 3: Using the Quadratic Formula
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Solve:
2𝑥² − 4𝑥 − 6 = 0
Solution:
$$a = 2, b = -4, c = -6 \Rightarrow x = \frac{-(-4) \pm \sqrt{(-4)² - 4(2)(-6)}}{2(2)} \ x = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4} \Rightarrow x = 3, x = -1$$
Detailed Explanation
In this example, we will solve the quadratic equation 2x² - 4x - 6 = 0 using the quadratic formula, which is:
x = (-b ± √(b² - 4ac)) / (2a).
- First, identify the coefficients a, b, and c in our equation. Here, a = 2, b = -4, and c = -6.
- Plug these values into the quadratic formula:
x = [-(-4) ± √((-4)² - 4(2)(-6))] / (2(2)). - Calculate the discriminant (the part under the square root):
- (-4)² = 16
- 4(2)(-6) = -48, thus 16 + 48 = 64.
- Knowing the discriminant, we can complete the formula as:
x = [4 ± √64] / 4. - Finally, √64 = 8, so we have two solutions:
x = (4 + 8)/4 = 3 and x = (4 - 8)/4 = -1.
Examples & Analogies
Using the quadratic formula is like using a universal key that fits many different types of locks—not just one. Even if your situation (or quadratic equation) changes, this versatile method will help you find the right solution every time, just like how the formula gives solutions for any quadratic equation.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Factoring: Rewriting a quadratic equation as a product of its factors to find solutions.
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Completing the Square: A method for manipulating a quadratic expression into a perfect square form to easily solve for x.
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Quadratic Formula: A universal formula to solve any quadratic equation.
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Discriminant: Important for understanding the number and type of solutions for quadratic equations.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Solve \( x^2 + 5x + 6 = 0 \) by factoring to find the roots \( x = -2, x = -3 \).
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Complete the square for \( x^2 + 4x + 1 = 0 \) to derive \( x = -2 \pm \sqrt{3} \).
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Use the quadratic formula on \( 2x^2 - 4x - 6 = 0 \) to determine \( x = 3 \) and \( x = -1 \).
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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To factor, just find what's neat, / Two numbers that add up to meet. / Then set each to zero in a row, / Your roots will show, watch them grow!
📖 Fascinating Stories
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Imagine a gardener (completing the square) adjusting a flower bed (quadratic) into a perfect circle (perfect square) to make sure every flower gets equal sunlight—this is how we adjust our equations!
🧠 Other Memory Gems
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For the quadratic formula, remember: \( 'B' is for 'bra' to hold the numbers together while calculating all roots at once!'
🎯 Super Acronyms
D.R.E.A.M. - Discriminant reveals each answer's nature
- Real roots
- Equal roots (zero)
- And imaginary roots (nonexistent solutions)!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Quadratic Function
Definition:
A polynomial function of degree 2, generally presented as \( f(x) = ax^2 + bx + c \).
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Term: Factoring
Definition:
The process of rewriting a polynomial as the product of its factors.
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Term: Completing the Square
Definition:
A method used to solve quadratic equations by converting them into a perfect square trinomial.
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Term: Quadratic Formula
Definition:
A formula providing the solutions to a quadratic equation in the form \( ax^2 + bx + c = 0 \).
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Term: Discriminant
Definition:
The part of the quadratic formula under the square root sign, \( b^2 - 4ac \), which indicates the nature of the roots.