Nature of Roots - Examples | Quadratic Equations | Don't Memorise
Summary
TLDRThis educational video script explains how to determine the nature of roots of a quadratic equation. It emphasizes the importance of the discriminant, b^2 - 4ac, in identifying whether roots are real and distinct, equal, or non-existent. The script guides through examples, showing how a discriminant of zero indicates equal real roots, a positive value suggests two distinct real roots, and a negative value means no real roots. It encourages viewers to solve and verify the nature of roots for three given quadratic equations.
Takeaways
- 📚 The nature of roots of a quadratic equation is determined by the discriminant, which is calculated as b^2 - 4ac.
- 🔍 If the discriminant is greater than zero, the equation has two distinct real roots.
- 🔄 If the discriminant equals zero, the equation has two equal real roots.
- 🚫 If the discriminant is less than zero, the equation has no real roots.
- 🧩 The first quadratic equation provided has a = 9, b = -12, and the discriminant is zero, indicating equal real roots.
- 🔢 For the first equation, the calculation of the discriminant is shown as (-12)^2 - 4 * 9 * 4 = 144 - 144 = 0.
- 📝 The second equation has a = 2, b = -9, and a positive discriminant, indicating two distinct real roots.
- 📉 The third equation's discriminant is negative, which means it has no real roots.
- 📚 The script encourages practice by solving the equations to verify the nature of the roots.
- 👨🏫 The explanation is instructional, guiding the learner through the process of determining the nature of roots for quadratic equations.
- 🎶 The script ends with a musical note, suggesting a conclusion or transition in the presentation.
Q & A
What is the general form of a quadratic equation?
-The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \).
What is the significance of the discriminant in a quadratic equation?
-The discriminant, denoted as \( \Delta = b^2 - 4ac \), determines the nature of the roots of a quadratic equation. If \( \Delta > 0 \), the equation has two distinct real roots; if \( \Delta = 0 \), it has two equal real roots; and if \( \Delta < 0 \), it has no real roots.
How do you calculate the discriminant for the quadratic equation \( 9x^2 - 12x + 4 = 0 \)?
-For the equation \( 9x^2 - 12x + 4 = 0 \), the discriminant is calculated as \( \Delta = (-12)^2 - 4 \times 9 \times 4 = 144 - 144 = 0 \).
What does a discriminant of zero indicate for the quadratic equation \( 9x^2 - 12x + 4 = 0 \)?
-A discriminant of zero indicates that the quadratic equation \( 9x^2 - 12x + 4 = 0 \) has two equal real roots.
What are the values of \( a \), \( b \), and \( c \) for the quadratic equation \( 2x^2 - 9x + 4 = 0 \)?
-For the equation \( 2x^2 - 9x + 4 = 0 \), the values are \( a = 2 \), \( b = -9 \), and \( c = 4 \).
How do you determine the nature of the roots for the quadratic equation \( 2x^2 - 9x + 4 = 0 \)?
-For the equation \( 2x^2 - 9x + 4 = 0 \), the discriminant is \( \Delta = (-9)^2 - 4 \times 2 \times 4 = 49 \). Since \( \Delta > 0 \), the equation has two distinct real roots.
What is the discriminant for the quadratic equation \( x^2 + 9x + 7 = 0 \)?
-For the equation \( x^2 + 9x + 7 = 0 \), the discriminant is \( \Delta = 9^2 - 4 \times 1 \times 7 = 81 - 28 = 53 \).
What does a positive discriminant signify for the roots of a quadratic equation?
-A positive discriminant signifies that the quadratic equation has two distinct real roots.
What are the steps to find the nature of the roots of a quadratic equation?
-The steps are: (1) Write down the quadratic equation in the form \( ax^2 + bx + c = 0 \), (2) Identify the values of \( a \), \( b \), and \( c \), (3) Calculate the discriminant \( \Delta = b^2 - 4ac \), and (4) Determine the nature of the roots based on the value of \( \Delta \).
Can a quadratic equation have complex roots?
-Yes, a quadratic equation can have complex roots if the discriminant is negative, as complex roots are not real numbers.
How can you verify the roots of a quadratic equation?
-You can verify the roots by substituting them back into the original equation and checking if both sides of the equation balance.
Outlines
📚 Understanding the Nature of Roots in Quadratics
This paragraph introduces the concept of determining the nature of roots for a quadratic equation, which is essential for solving them. It emphasizes that the focus is on the nature of the roots, not their actual values. The explanation revolves around the discriminant (b^2 - 4ac), which dictates the roots' nature: two distinct real roots for a positive discriminant, equal real roots for zero, and no real roots for a negative one. The paragraph also guides through an example, finding the values of a, b, and c, and calculating the discriminant to conclude the roots' nature.
Mindmap
Keywords
💡Quadratic Equation
💡Roots
💡Discriminant
💡Nature of Roots
💡Real Roots
💡Distinct Roots
💡Equal Roots
💡No Real Roots
💡Coefficients
💡Practice
💡Verification
Highlights
The nature of roots of a quadratic equation depends on the discriminant value.
A quadratic equation in the form of ax^2 + bx + c = 0 has roots determined by a specific formula.
A discriminant value greater than zero indicates two distinct real roots.
A discriminant of zero signifies equal real roots for the quadratic equation.
A negative discriminant means the quadratic equation has no real roots.
The first quadratic equation provided has a = 9, b = -12, resulting in a discriminant of 0.
The equation with a discriminant of 0 has two equal real roots.
The second equation has a = 2, b = -9, and c = 4, leading to a positive discriminant.
A positive discriminant in the second equation implies two distinct real roots.
The third equation has a discriminant of negative seven, indicating no real roots.
The process of finding the nature of roots involves comparing the given equation to the general form.
Calculating b^2 - 4ac is essential to determine the discriminant's value.
The discriminant's sign is the key to understanding the roots' nature.
Practical application of the discriminant is demonstrated through solving example equations.
The transcript encourages pausing to verify the roots' nature by solving the equations.
Solving the equations provides a good opportunity for practice in understanding the discriminant.
The transcript concludes with a reminder of the importance of the discriminant in quadratic equations.
Transcripts
we need to find the nature of roots of
the following quadratic equation notice
that we've been asked only for the
nature of the roots and not values of
the root the first thing we need to know
is that for the quadratic equation ax
squared plus BX plus C equals 0 the
roots are given by this formula and the
nature of the roots depends on this
value which is called the discriminant
if it's greater than zero and the
quadratic equation will have two
distinct real roots if it's equal to
zero then the quadratic equation will
have equal real roots and if it's less
than zero then the quadratic equation
will have no real roots keep this idea
in mind let me solve the first one for
you the first thing we need to do is
find the values of a B and C comparing
the first equation with the general form
we get the value of a at 9 B as negative
12 and C as for now it's easy to
calculate the value of b squared minus
4ac it will equal negative 12 squared
minus 4 times 9 times 4 negative 12
squared is 144 and 4 times 9 times 4 is
also 144 so we get the value of the
discriminant as 0 what does this tell us
it tells us that this quadratic equation
has to equal and real roots try pausing
for the roots and verify you answer now
I want you to try finding the nature of
the roots of the second question
comparing the second equation with the
general form we get the value of a as to
be as negative 9 and C at 4 what will be
the value of b squared minus 4ac then it
will equal 49 which is positive this
quadratic equation will have two
distinct real roots as its discriminant
is positive and you should be able to
solve the third one easily we get the
value of the discriminant at negative
seven this quadratic equation will have
no real rules you should try solving the
above equations to verify your answer
and it's a good chance for practice
[Music]
Voir Plus de Vidéos Connexes
MATH9 DISCRIMINANT and NATURE OF ROOTS of quadratic equation #math9 #discriminant #natureofroots
Solving Quadratic Equations by Quadratic Formula | Not A Perfect Square | Part 2 |
Polinomial (Bagian 5) - Cara Menentukan Akar-akar Persamaan Polinomial
Sum and Product of the Roots of Quadratic Equation - Finding the Quadratic Equation
completing the square with complex roots (KristaKingMath)
MATH 9 - Solving Equations Transformable to Quadratic Equation Including Rational Algebraic Equation
5.0 / 5 (0 votes)