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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Today we're going to learn how to solve quadratic equations by factorization. Can anyone tell me what factorization means in this context?
It means breaking down the equation into simpler parts that we can solve.
Exactly! For example, if we have the equation x² - 7x + 12 = 0, how would we factor it?
We need two numbers that multiply to 12 and add up to -7, those are -3 and -4.
Great job! So we can write this as (x - 3)(x - 4) = 0. What do we do next?
We set each factor equal to zero and solve for x, so x = 3 and x = 4.
Right! Excellent work, everyone. Remember, FACTOR stands for Finding and Creating Terms to Obtain Roots. Let's keep that in mind as we move on.
Now, let's tackle another quadratic equation: 3x² + 5x - 2 = 0. Who can remind us of the quadratic formula?
It's x = (-b ± √(b² - 4ac)) / 2a!
Perfect! Here, a is 3, b is 5, and c is -2. What’s the first step?
Plugging the values into the formula!
That’s right! So we calculate x = (−5 ± √(5² - 4 × 3 × -2))/ (2 × 3). What do we find?
We get x = (−5 ± √(25 + 24)) / 6, which simplifies to x = (−5 ± √49) / 6.
Exactly! And what are the roots?
The roots are x = 1/2 and x = -3!
Well done, everyone! Remember, applying the formula can help us find roots when factorization is too complex. The acronym ROOTS helps us remember: Right Order of Terms Solves!
Next, we’ll learn about completing the square with another example. Can one of you share why this method is helpful for us?
It helps find the vertex of the parabola!
Exactly! Now, consider the equation x² + 8x + 10 = 0. What’s our first step?
We should shift 10 to the other side of the equation first, so x² + 8x = -10.
Right! Now, what’s next?
We take half of 8, which is 4, and then square it to get 16.
Great! Add 16 to both sides. What do we get?
We have (x + 4)² = 6.
Perfect! And then, how do we solve for x?
We take the square root of both sides, so x + 4 = ±√6, leading to x = -4 ± √6.
Excellent work, everyone! The memory aid for this method is COMPLETE: Create One Missing Part; Let Equations transmit answers!
Let's discuss graphing our quadratic function: f(x) = -x² + 6x - 5. What’s the first thing we should identify?
We need to find the vertex!
Exactly! Can someone remind us how to find the vertex?
Use the formula x = -b/2a. Here, a is -1 and b is 6.
Fantastic! So what do we calculate?
x = -6 / (2 × -1) gives us x = 3.
Great! Now, what’s the y-value at x = 3?
f(3) = -(3)² + 6(3) - 5, which equals 4!
Perfect! The vertex is at (3, 4). And what do we know about the graph since a is negative?
It opens downwards, creating a maximum point!
Exactly! Quadratic functions can peak or dip, forming a U-shape or an inverted U-shape. Remember the idea of PEAK: Parabolas Emit Awesomeness, Keep Upward or Downward!
Finally, let's apply what we've learned to a real-world context. According to projectile motion, the height of an object can be modeled by a quadratic function. What’s a familiar example?
Like a ball thrown in the air!
Exactly! Now, consider the function h(t) = -5t² + 20t + 1. What are we trying to determine?
The maximum height of the ball and when it hits the ground.
Right! How would we find the maximum height?
We find the vertex of the quadratic; that gives the maximum height.
Well done! Remember, to find when it hits the ground, we solve h(t) = 0. The method learned today in class will apply in many real-life scenarios. Keep using the acronym REAL: Real-life Equations Apply Learning!
To graph quadratic with grace and ease, find the vertex, it’s sure to please.
Imagine a ball thrown high, reaching its max, then falling from the sky, a parabolic path, like a bird's flight, the vertex is the peak, a beautiful sight!
Remember as we graph: Vertex Equals x-axis Reach. (V = x-axis)
ROOT: Real Outcomes Originate from Terms.
Term
What is a quadratic function?
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Term
How do you find the vertex?
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Term
What is the significance of the discriminant?
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A polynomial function of degree 2, typically in the form f(x) = ax² + bx + c.
The highest or lowest point of a parabola, determined by the formula x = -b/(2a).
A vertical line that divides the parabola into two mirror-image halves, represented by x = -b/(2a).
The value of b² - 4ac which indicates the number of solutions to a quadratic equation.
This section presents practice questions aimed at reinforcing understanding of quadratic functions and their applications.
The Practice Questions section includes various questions that require students to apply their knowledge of quadratic functions, such as solving equations via different methods, analyzing graphs, and interpreting real-life scenarios modeled by quadratics.
In this section, students are provided with a series of practice questions that explore different aspects of quadratic functions. These questions cover essential topics, including solving quadratic equations using factoring, the quadratic formula, and completing the square. The section emphasizes practical applications, as seen through a word problem related to projectile motion. By engaging with these practice questions, students will reinforce their understanding of key concepts such as the structure of quadratic equations, transformations, vertex identification, graphing parabolas, and the significance of the discriminant in determining the nature of the roots.