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Applications of the Discriminant>
Applications of the Discriminant 2Applications of the Discriminant 2
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Question 1 of 6
1. Question
Using the discriminant, find the nature of the roots of the function:`x^2-6x-4=0`Hint
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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
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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
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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
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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
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Excellent!
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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
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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
- Sum & Product of Roots 1
- Sum & Product of Roots 2
- Sum & Product of Roots 3
- Sum & Product of Roots 4
- Solving Equations by Factoring 1
- Solving Equations Using 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
- Solving Reducible Equations