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Questions and Answers
What is the vertex form of the quadratic equation \(2x^2 - 4x - 6 = 0\)?
What is the vertex form of the quadratic equation \(2x^2 - 4x - 6 = 0\)?
For the quadratic equation \(3x^2 + 6x + 4 = 0\), what is the value of the discriminant?
For the quadratic equation \(3x^2 + 6x + 4 = 0\), what is the value of the discriminant?
What is the vertex of the parabola described by the equation \(x^2 + 4x + 5 = 0\)?
What is the vertex of the parabola described by the equation \(x^2 + 4x + 5 = 0\)?
If a ball is thrown upwards and follows a path modeled by the quadratic equation \(3t^2 + 6t + 4 = 0\), what does the coefficient of t squared (3) represent?
If a ball is thrown upwards and follows a path modeled by the quadratic equation \(3t^2 + 6t + 4 = 0\), what does the coefficient of t squared (3) represent?
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What is the graph of the quadratic equation \(x^2 - x - 6 = 0\) shaped as?
What is the graph of the quadratic equation \(x^2 - x - 6 = 0\) shaped as?
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In a word problem involving a quadratic equation, when calculating the vertex form, why is it important to complete the square?
In a word problem involving a quadratic equation, when calculating the vertex form, why is it important to complete the square?
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What does the discriminant of a quadratic equation help determine?
What does the discriminant of a quadratic equation help determine?
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How can you find the vertex of a quadratic equation?
How can you find the vertex of a quadratic equation?
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In what scenario does a quadratic equation have two distinct real roots?
In what scenario does a quadratic equation have two distinct real roots?
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What does the y-intercept represent on a graph of a quadratic equation?
What does the y-intercept represent on a graph of a quadratic equation?
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How do you determine the number of x-intercepts on a graph of a quadratic equation?
How do you determine the number of x-intercepts on a graph of a quadratic equation?
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In a word problem, what does the vertex of a quadratic equation represent?
In a word problem, what does the vertex of a quadratic equation represent?
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Study Notes
Quadratic Equations
Vertex Form
A quadratic equation is a polynomial function of degree 2. It can be written in the general form:
[ ax^2 + bx + c = 0 ]
where (a,b,) and (c) are constants. If the equation is already in the form of (ax^2 + bx + c = 0), it is in standard form. If the quadratic equation is in the form (a(x - h)^2 + k = 0), it is in vertex form.
Vertex Formula
To convert a quadratic equation from standard form to vertex form, follow these steps:
- Complete the square of the quadratic term, (ax^2).
- Add and subtract the square of half of the linear term, (\frac{b}{2a}).
- Factor out the perfect square, (a).
- Move the constant term, (c), to the right side of the equation.
For example, consider the quadratic equation (x^2 + 2x - 7 = 0). To find the vertex form, we complete the square:
[ \begin{align*} x^2 + 2x - 7 &= 0 \ \left(x + 1\right)^2 - 1 &= 0 \ \left(x + 1\right)^2 &= 1 \ x^2 + 2x + 1 &= 1 \ \left(x^2 + 2x + 1\right) - 1 &= 0 \ \left(x + 1\right)^2 - 1 &= 0 \ \left(x - \frac{1}{2}\right)^2 &= \frac{1}{4} \end{align*} ]
The vertex form is (\left(x - \frac{1}{2}\right)^2 = \frac{1}{4}).
Discriminant
The discriminant, denoted as (\Delta), is the value found by taking the square of the coefficient of the linear term, (b), and subtracting 4 times the product of the coefficients of the quadratic term, (a), and the constant term, (c):
[ \Delta = b^2 - 4ac ]
The discriminant can be used to determine the nature of the roots of the quadratic equation. If (\Delta > 0), the equation has two distinct real roots. If (\Delta = 0), the equation has one real root or a repeated root. If (\Delta < 0), the equation has two complex roots.
Graphing
To graph a quadratic equation, follow these steps:
- Find the vertex by evaluating the formula (x = \frac{-b}{2a}).
- Find the (x)-intercepts by setting (y) to 0 and solving for (x).
- Find the (y)-intercept by evaluating the equation at (x = 0).
- Plot the vertex and the (x)-intercepts, and then use the symmetry about the (y)-axis to complete the graph.
Word Problems
Word problems in quadratic equations can be solved using the same steps as for standard form equations. However, it is essential to identify the terms and constants in the equation and determine their relationships to the variables in the problem.
For example, consider the problem: "A ball is thrown upwards with an initial velocity of 20 meters per second. Its height above the ground is given by the equation (h = -16t^2 + 20t), where (h) is the height in meters and (t) is the time in seconds. How long does it take for the ball to reach its maximum height?"
To solve this problem, we need to determine the vertex of the quadratic equation, which represents the maximum height of the ball. The vertex form of the equation is (h = -16(t - 1)^2 + 20(t - 1)). By evaluating the vertex formula, we find that the maximum height occurs at (t = 1) seconds. Therefore, it takes 1 second for the ball to reach its maximum height.
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Description
Learn about quadratic equations in vertex form, how to convert standard form to vertex form, the discriminant to determine nature of roots, graphing techniques including finding the vertex and intercepts, and solving word problems involving quadratic equations. Practice your understanding of quadratic functions.
