Writing Quadratic Equations In Standard Form Given The Solution
TLDRThe video script provides a detailed tutorial on constructing quadratic equations from given roots. It demonstrates converting roots into factored form, then expanding to standard form. The script covers various scenarios, including equations with real and imaginary roots, and even cases with three given points. Techniques for simplifying the process, such as using conjugates and avoiding complex multiplication, are also discussed. The tutorial concludes with a method to derive a quadratic equation from three points, solving for coefficients using elimination.
Takeaways
- π The script explains how to form a quadratic equation from given roots.
- π It demonstrates converting the roots of a quadratic equation into a factored form and then expanding it to standard form.
- π The example of roots 2 and -3 is used to derive the quadratic equation x^2 + x - 6 = 0.
- π€ The script discusses the process of dealing with fractions and how to eliminate them by multiplying through by a common denominator.
- π For roots like 3 and 2/3, the script shows how to find the corresponding factors and then expand to form the quadratic equation 3x^2 - 11x + 6 = 0.
- π The concept of radicals and imaginary numbers in roots is introduced, with examples of how to form quadratic equations from them.
- 𧩠It explains that when given one root, the other must be its opposite, and how to use this to form the equation.
- π’ The script provides a method for finding a quadratic equation when given three points, using substitution to find the constant term.
- π It shows how to use the method of elimination to solve for the coefficients of the quadratic equation given multiple points.
- π The final quadratic equation derived from the points is -6x^2 + 9x - 4 = 0, and the script verifies it by plugging in the given points.
- π The script emphasizes the importance of checking the work by substituting the points back into the derived equation to ensure correctness.
Q & A
What is the standard form of a quadratic equation?
-The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).
How can you write a quadratic equation in factored form if the roots are given as 2 and -3?
-If the roots are 2 and -3, the factored form of the quadratic equation is \( (x - 2)(x + 3) \).
What is the result of the quadratic equation in standard form when the roots are 2 and -3?
-When the roots are 2 and -3, the quadratic equation in standard form is \( x^2 + x - 6 = 0 \).
How do you find the quadratic equation when given two solutions, 3 and 2/3?
-To find the quadratic equation, first write the factors as \( x - 3 \) and \( x - \frac{2}{3} \), then multiply these factors to get \( x^2 - \frac{11}{3}x + 2 = 0 \).
What is the relationship between the roots of a quadratic equation when dealing with radicals?
-When dealing with radicals, the roots come in pairs. If one root is \( \sqrt{3} \), the other must be \( -\sqrt{3} \).
How can you write a quadratic equation with one given real root and one given imaginary root?
-If one root is \( 3i \) and the other is \( -3i \), the quadratic equation in factored form is \( (x - 3i)(x + 3i) \), which simplifies to \( x^2 + 9 = 0 \).
What is the conjugate of a complex number, and how is it used in forming a quadratic equation?
-The conjugate of a complex number \( a + bi \) is \( a - bi \). It is used to form a quadratic equation in factored form as \( (x - (a + bi))(x - (a - bi)) \).
How can you simplify the process of finding a quadratic equation given two complex roots?
-You can use the conjugate of the complex roots and apply the technique of squaring the binomial plus the product of the imaginary parts to simplify the process.
What is the method to find a quadratic equation given three points on the graph?
-Use the points to create a system of equations based on the standard form \( ax^2 + bx + c \). Solve the system to find the values of \( a \), \( b \), and \( c \).
How do you verify that a quadratic equation fits three given points?
-Plug each point into the quadratic equation and check if the left side equals the y-value of the point, confirming that the equation fits all three points.
Outlines
π Writing Quadratic Equations from Roots
This paragraph explains the process of deriving a quadratic equation from its roots. It starts with a basic example where the roots are 2 and -3, and demonstrates how to write the equation in factored form as (x - 2)(x + 3) and then expands it to x^2 + x - 6. The explanation continues with the case where the roots are fractions, such as 3/2, and how to manipulate the equation to eliminate the fraction and arrive at a quadratic equation. The paragraph also covers the scenario where only one root is given, such as 3, and how to account for the paired root, -3, to form the equation x^2 - 3. The discussion extends to complex roots, showing how to handle them in a similar manner to form the quadratic equation x^2 + 9. Finally, it touches on the conjugate roots, like 2 + 3i and 2 - 3i, and how they lead to the equation x^2 + 13.
π Advanced Factoring Techniques
This section delves into advanced factoring techniques for quadratic equations, particularly when dealing with complex numbers. It begins by illustrating the process of multiplying two binomials with complex roots, such as (x - 2 + 3i) and (x - 2 - 3i), and explains the shortcut method that simplifies the expansion to (x - 2)^2 + 9i^2, resulting in the quadratic equation x^2 - 4x + 13 = 0. The paragraph also introduces an alternative, more efficient technique that leverages the identity (x - a)^2 + b^2 to quickly arrive at the same quadratic equation. Additionally, it discusses the scenario where three points are given, and how to use these points to determine the coefficients a, b, and c of the standard form quadratic equation, ax^2 + bx + c = 0, through a series of equations and elimination.
π Constructing Quadratic Equations from Points
The final paragraph focuses on constructing a quadratic equation in standard form given three points on the graph of the equation. It outlines the steps to find the values of a, b, and c by substituting the x and y values of each point into the general form of a quadratic equation. The process involves solving a system of equations derived from the points to isolate the coefficients. The paragraph demonstrates the elimination method to find the values of a and b, and then uses these values to verify the correctness of the quadratic equation by substituting the original points back into the equation. The verification process confirms that the derived equation accurately represents the given points, ensuring the correctness of the solution.
Mindmap
Keywords
π‘Quadratic Equation
π‘Roots
π‘Standard Form
π‘Factored Form
π‘FOIL Method
π‘Conjugate
π‘Imaginary Numbers
π‘Radicals
π‘Three Points
π‘Elimination Method
Highlights
Writing a quadratic equation in standard form given its roots.
Quadratic equation factored form using roots 2 and -3.
Foil method to expand the factored form into a quadratic equation.
Quadratic equation x^2 + x - 6 derived from given roots.
Adjusting factors for non-integer roots like 3 and 2/3.
Eliminating fractions by multiplying through by a common denominator.
Quadratic equation derived from roots 3 and 2/3.
Understanding that radicals come in pairs in quadratic equations.
Quadratic equation x^2 - 3 from a single positive root.
Handling imaginary roots and their conjugates in quadratic equations.
Quadratic equation x^2 + 9 from imaginary roots 3i and -3i.
Conjugate pairs for complex roots in quadratic equations.
Two methods for deriving quadratic equations from complex roots.
Shortcut method for foiling trinomials in quadratic equations.
Quadratic equation x^2 - 4x + 13 from complex roots 2 + 3i and 2 - 3i.
Using three given points to construct a quadratic equation.
Finding the constant term c from the y-intercept.
Solving for coefficients a and b using a system of equations.
Verification of the derived quadratic equation with given points.
Final quadratic equation -6x^2 + 9x - 4 matches all given points.
Transcripts
Browse More Related Video
Writing Quadratic Equations In Vertex Form & Standard Form Given 3 Points
How to solve a quadratic equation with Casio fx-991MS edition 2 scientific calculator
How To Find The Real & Imaginary Solutions of Polynomial Equations
Factorizing Using Casio 570MS Calculator
How To Find The Vertex of a Parabola - Standard Form, Factored & Vertex Form
How To Solve Systems of Nonlinear Equations
5.0 / 5 (0 votes)
Thanks for rating: