Topics
>
Algebra 1>
Quadratic Equations and Functions>
Applications of the Discriminant>
Applications of the Discriminant 2Applications of the Discriminant 2
Try VividMath Premium to unlock full access
Time limit: 0
Quiz summary
0 of 6 questions completed
Questions:
- 1
- 2
- 3
- 4
- 5
- 6
Information
–
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading...
You must sign in or sign up to start the quiz.
You have to finish following quiz, to start this quiz:
Loading...
- 1
- 2
- 3
- 4
- 5
- 6
- Answered
- Review
-
Question 1 of 6
1. Question
Using the discriminant, find the nature of the roots of the function:`x^2-6x-4=0`Hint
Help VideoCorrect
Excellent!
Incorrect
Nature of the Roots Discriminant (`Delta`) Two real roots `Delta``>``0` One real root `Delta=0` No real roots `Delta``<``0` Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`x^2-6x-4=0``a=1` `b=-6` `c=-4``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{(-6)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(-4)}$$ Substitute values `=` `36+16` `=` `52` This is a positive value, which means `Delta``>``0`Therefore, the function has Two real rootsTwo real roots -
Question 2 of 6
2. Question
Identify which values of `k` will make the function below have no real roots`kx^2+4x+1=0`Hint
Help VideoCorrect
Great Work!
Incorrect
Nature of the Roots Discriminant (`Delta`) Two real roots `Delta``>``0` One real root `Delta=0` No real roots `Delta``<``0` Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`kx^2+4x+1=0``a=k` `b=4` `c=1``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{4}}^2-4\color{#00880A}{(k)}\color{#007DDC}{(1)}$$ Substitute values `=` `16-4k` Remember that for a function to have one real root, `Delta``<``0`Substitute the `Delta` computed previously, and then solve for `k``Delta` `<` `0` `16-4k` `<` `0` `16-4k` `-16` `<` `0` `-16` Subtract `16` from both sides `-4k` `<` `-16` `-4k``divide-4` `<` `-16``divide-4` Divide both sides by `-4` `k` `>` `4` An inequality flips when divided by a negative `k``>``4` -
Question 3 of 6
3. Question
Identify which values of `m` will make the function below have real roots`x^2+(m+2)x+1=0`Hint
Help VideoCorrect
Fantastic!
Incorrect
Nature of the Roots Discriminant (`Delta`) Two real roots `Delta``>``0` One real root `Delta=0` No real roots `Delta``<``0` Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`x^2+(m+2)x+1=0``a=1` `b=m+2` `c=1``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{(m+2)}}^2-4\color{#00880A}{(1)}\color{#007DDC}{(1)}$$ Substitute values `=` `m^2+4m+4-4` `=` `m^2+4m` A function that has real roots can have either one or two real roots, hence `Delta``≥``0`Substitute the `Delta` computed previously, and then solve for `m``Delta` `≥` `0` `m^2+4m` `≥` `0` `m(m+4)` `≥` `0` `m=0` `m=-4` To determine which region around `m=0` and `m=-4` would be included, plot these points and make a rough sketch of `m^2+4m`Replace the `x` axis with `m` axis and draw an upward parabola since `1` is positive
Remember that `Delta` must be positiveTherefore, `m``≤``-4` and `m``≥``0``m``≤``-4` and `m``≥``0` -
Question 4 of 6
4. Question
Identify which values of `m` will make the function below have one real root`2x^2+mx+m-2=0`- `m=` (4)
Hint
Help VideoCorrect
Well Done!
Incorrect
Nature of the Roots Discriminant (`Delta`) Two real roots `Delta``>``0` One real root `Delta=0` No real roots `Delta``<``0` Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`2x^2+mx+m-2=0``a=2` `b=m` `c=m-2``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{m}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(m-2)}$$ Substitute values `=` `m^2-8m+16` Remember that the function must have one real root, hence `Delta=0`Substitute the `Delta` computed previously, and then solve for `m``Delta` `=` `0` `m^2-8m+16` `=` `0` [insert cross method image with two `m`’s on the left and two `-4`’s on the right]`(m-4)(m-4)` `=` `0` `m=4` `m=4` -
Question 5 of 6
5. Question
Identify which values of `k` will make the function below have no real roots`kx^2-4x+k=0`Hint
Help VideoCorrect
Excellent!
Incorrect
Nature of the Roots Discriminant (`Delta`) Two real roots `Delta``>``0` One real root `Delta=0` No real roots `Delta``<``0` Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`kx^2-4x+k=0``a=k` `b=-4` `c=k``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{(-4)}}^2-4\color{#00880A}{(k)}\color{#007DDC}{(k)}$$ Substitute values `=` `16-4k^2` Remember that the function must have no real roots, hence `Delta``<``0`Substitute the `Delta` computed previously, and then solve for `k``Delta` `<` `0` `16-4k^2` `<` `0` `4(4-k^2)` `<` `0` `4(2-k)(2+k)` `<` `0` `k=-2` `k=2` To determine which region around `k=-2` and `k=2` would be included, plot these points and make a rough sketch of `16-4k^2`Replace the `x` axis with `k` axis and draw a downward parabola since `-4` is negative
Remember that `Delta` must be negativeTherefore, `k``<``-2` and `k``>``2``k``<``-2` and `k``>``2` -
Question 6 of 6
6. Question
Identify which values of `m` will make the function below have two real roots`2x^2+mx+8=0`Hint
Help VideoCorrect
Keep Going!
Incorrect
Nature of the Roots Discriminant (`Delta`) Two real roots `Delta``>``0` One real root `Delta=0` No real roots `Delta``<``0` Discriminant Formula
$$\Delta={\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$First, compute for the discriminant`2x^2+mx+8=0``a=2` `b=m` `c=8``Delta` `=` $${\color{#9a00c7}{b}}^2-4\color{#00880A}{a}\color{#007DDC}{c}$$ Discriminant Formula `=` $${\color{#9a00c7}{m}}^2-4\color{#00880A}{(2)}\color{#007DDC}{(8)}$$ Substitute values `=` `m^2-64` Remember that the function must have two real roots, hence `Delta``>``0`Substitute the `Delta` computed previously, and then solve for `m``Delta` `>` `0` `m^2-64` `>` `0` `(m+8)(m-8)` `>` `0` `m=-8` `m=8` To determine which region around `m=-8` and `k=8` would be included, plot these points and make a rough sketch of `m^2-64`Replace the `x` axis with `m` axis and draw an upward parabola since `1` is positive
Remember that `Delta` must be positiveTherefore, `m``<``-8` and `m``>``8``m``<``-8` and `m``>``8`
Quizzes
- Solve Quadratics by Factoring
- The Quadratic Formula
- Completing the Square 1
- Completing the Square 2
- Intro to Quadratic Functions (Parabolas) 1
- Intro to Quadratic Functions (Parabolas) 2
- Intro to Quadratic Functions (Parabolas) 3
- Graph Quadratic Functions in Standard Form 1
- Graph Quadratic Functions in Standard Form 2
- Graph Quadratic Functions by Completing the Square
- Graph Quadratic Functions in Vertex Form
- Write a Quadratic Equation from the Graph
- Write a Quadratic Equation Given the Vertex and Another Point
- Quadratic Inequalities 1
- Quadratic Inequalities 2
- Quadratics Word Problems 1
- Quadratics Word Problems 2
- Quadratic Identities
- Graphing Quadratics Using the Discriminant
- Positive and Negative Definite
- Applications of the Discriminant 1
- Applications of the Discriminant 2
- Combining Methods for Solving Quadratic Equations