Podcast
Questions and Answers
Which of the following is the correct formula for calculating the discriminant of a quadratic equation in the form $ax^2 + bx + c = 0$?
Which of the following is the correct formula for calculating the discriminant of a quadratic equation in the form $ax^2 + bx + c = 0$?
- $b^2 + 4ac$
- $b^2 - 4ac$ (correct)
- $-b +\sqrt{b^2 - 4ac}$
- $\sqrt{b^2 - 4ac}$
A discriminant of zero indicates that the quadratic equation has two distinct real roots.
A discriminant of zero indicates that the quadratic equation has two distinct real roots.
False (B)
If the discriminant of a quadratic equation is positive, how many real roots does the equation have?
If the discriminant of a quadratic equation is positive, how many real roots does the equation have?
Two
If the discriminant is a perfect square, the roots of the quadratic equation are ______.
If the discriminant is a perfect square, the roots of the quadratic equation are ______.
Match the discriminant value to the correct statement about the nature of roots:
Match the discriminant value to the correct statement about the nature of roots:
Which statement is true when the discriminant of a quadratic equation is negative?
Which statement is true when the discriminant of a quadratic equation is negative?
If a line intersects a parabola at one point, the discriminant of the resulting quadratic equation will be negative.
If a line intersects a parabola at one point, the discriminant of the resulting quadratic equation will be negative.
To solve simultaneous equations involving a line and a parabola, what is the first step after setting up the equations?
To solve simultaneous equations involving a line and a parabola, what is the first step after setting up the equations?
The points of intersection between a line and a parabola are found by solving the simultaneous equations and determining the ______ that satisfy both equations.
The points of intersection between a line and a parabola are found by solving the simultaneous equations and determining the ______ that satisfy both equations.
Match the number of intersection points between a line and a parabola with the corresponding number of solutions to the simultaneous equations:
Match the number of intersection points between a line and a parabola with the corresponding number of solutions to the simultaneous equations:
What does it mean if the discriminant of the equation formed after substituting a linear equation into a quadratic equation is zero?
What does it mean if the discriminant of the equation formed after substituting a linear equation into a quadratic equation is zero?
If substituting a linear equation into a quadratic equation results in a discriminant less than zero, then the line intersects the parabola at two points.
If substituting a linear equation into a quadratic equation results in a discriminant less than zero, then the line intersects the parabola at two points.
When solving simultaneous equations of a line and a parabola, what variable is eliminated by substituting the linear equation into the quadratic equation?
When solving simultaneous equations of a line and a parabola, what variable is eliminated by substituting the linear equation into the quadratic equation?
If the roots of the resulting quadratic equation, after substitution, are irrational, the intersection points' coordinates will also be ______.
If the roots of the resulting quadratic equation, after substitution, are irrational, the intersection points' coordinates will also be ______.
Match each scenario with the correct number of solutions when solving a line and parabola simultaneously.
Match each scenario with the correct number of solutions when solving a line and parabola simultaneously.
What is the purpose of using the discriminant after substituting a linear equation into a quadratic equation?
What is the purpose of using the discriminant after substituting a linear equation into a quadratic equation?
If the discriminant of the combined equation is positive, the line and parabola are parallel and do not intersect.
If the discriminant of the combined equation is positive, the line and parabola are parallel and do not intersect.
When solving simultaneous equations of a line and a parabola, if the discriminant is zero, what term is used to describe the line's relationship to the parabola?
When solving simultaneous equations of a line and a parabola, if the discriminant is zero, what term is used to describe the line's relationship to the parabola?
If the line is represented by $y = mx + c$ and the parabola by $y = ax^2 + bx + d$, you ______ $mx + c$ into the $y$ of the quadratic equation to find the points of intersection.
If the line is represented by $y = mx + c$ and the parabola by $y = ax^2 + bx + d$, you ______ $mx + c$ into the $y$ of the quadratic equation to find the points of intersection.
Match each value of 'a' with the correct shape and orientation of its parabola.
Match each value of 'a' with the correct shape and orientation of its parabola.
If the parabola $y=ax^2+bx+c$ opens downward, what can you say about the value of $a$?
If the parabola $y=ax^2+bx+c$ opens downward, what can you say about the value of $a$?
If the equation has no point of intersection, the line makes no contact with the parabola.
If the equation has no point of intersection, the line makes no contact with the parabola.
What happens when there is 1 point of intersection between a non-vertical line and a parabola.
What happens when there is 1 point of intersection between a non-vertical line and a parabola.
When there are 2 points of intersection, the line cuts through the ______ at these points.
When there are 2 points of intersection, the line cuts through the ______ at these points.
Match the root definition with its type.
Match the root definition with its type.
What can you use to solve a quadratic equation to find the number and nature of your roots?
What can you use to solve a quadratic equation to find the number and nature of your roots?
What does the graph look like if ᐃ=0?
What does the graph look like if ᐃ=0?
What word is used when ᐃ < 0? ______ definitive
What word is used when ᐃ < 0? ______ definitive
Math the two types of roots
Math the two types of roots
If ᐃ > 0 there are no x-intercepts?
If ᐃ > 0 there are no x-intercepts?
Flashcards
What is the discriminant?
What is the discriminant?
The discriminant is b² - 4ac (from the quadratic equation ax² + bx + c = 0).
Δ > 0: Number of roots?
Δ > 0: Number of roots?
If the discriminant (Δ) is greater than 0, there are two distinct real roots.
Δ = 0: Number of Roots?
Δ = 0: Number of Roots?
If the discriminant (Δ) is equal to 0, there is exactly one real root (a repeated root).
Δ < 0: Number of Roots?
Δ < 0: Number of Roots?
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Rational vs. Irrational Roots?
Rational vs. Irrational Roots?
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Solving Line-Parabola Intersections
Solving Line-Parabola Intersections
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Tangent Line
Tangent Line
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Discriminant and Intersection Points
Discriminant and Intersection Points
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Study Notes
- The lesson will cover the discriminant and solving simultaneous equations.
- The learning intention is to consolidate understanding of the role of the discriminant.
- To learn how to solve simultaneous equations where a line intersects a parabola.
Discriminant Formula Practice:
- Determine the formula for the discriminant for the equation y = 3x² + 2x – 3.
- Calculate the discriminant for the equation y = 3x² + 2x – 3.
- State the number of solutions based on the discriminant calculation for y = 3x² + 2x – 3.
- State the coordinates of the y-intercept for the equation y = 3x² + 2x – 3.
- State the x-coordinate of the turning point for the equation y = 3x² + 2x – 3.
The Role of the Discriminant:
- The discriminant determines the number of roots a quadratic equation has.
- If the discriminant is a square number, then the roots are rational.
- If not, the two solutions are irrational and contain surds.
- If the discriminant is equal to 0, there is one root, and the quadratic is a perfect square.
- Technology: You can solve or sketch the equation on a calculator to find the number and nature of the roots.
- If Δ=0, there is only one x-intercept and the graph touches the axis at the turning point.
- If Δ>0, there are two x-intercepts and the graph cuts through the axis at two different points.
- If Δ<0, there are no x-intercepts.
Intersection of Lines and Parabolas:
- The possible intersections of a straight line and a parabola are 0, 1, and 2.
- If there is no point of intersection, the line makes no contact with the parabola.
- If there is 1 point of intersection, a non-vertical line is a tangent line to the parabola, touching the parabola at that one point of contact.
- If there are 2 points of intersection, the line cuts through the parabola at these points.
- Simultaneous equations can be solved to find the points of intersection of a line and a parabola.
- To do this, substitute the linear equation into the quadratic equation, to form one equation.
- The discriminant can be used to determine the number of solutions and therefore the number of intersections.
- Technology: The equations can be sketched on a graphics calculator to find the intersection point's coordinates.
Homework: Exercise 2.5
- Mild: Q1a, Q3a, Q6a, Q10-11, Q14a
- Med: Q1b, Q3b, Q6d, Q12ab, Q14b
- Spicy: Q1d, Q3d, Q6f, Q12cd, Q15
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