Which Function Has a Negative Discriminant Value
Which Function Has a Negative Discriminant Value.
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Contents
 0.1 Pre Requisite Cognition
 0.1.1 Pre Req i : What does the graph of a quadratic equation look like:
 0.1.2 Pre Req two : What is the solution of a quadratic equation:
 0.1.3 What does the discriminant look like?
 0.1.4 What is the discriminant anyway?
 0.1.5 What is the formula for the Discriminant?
 0.1.6 What does this formula tell usa?
 0.2 Nature of the Solutions
 0.3 Case
 0.4 Practice Problems
 1 Which Function Has a Negative Discriminant Value
Pre Requisite Cognition
To understand what the discriminant does, it’southward important that y’all have a good understanding of:
Pre Req i : What does the graph of a quadratic equation look like:
Pre Req two : What is the solution of a quadratic equation:
Reply
The solution can be idea of in two dissimilar means.

Algebraically, the solution occurs when y = 0. Then the solution is where
$$y =\cherryred ax^two + \blue bx + \color{greenish} c $$
becomes
$$0 =\red ax^2 + \bluish bx + \color{green} c $$. 
Graphically, since y = 0 is the xaxis, the solution is where the parabola intercepts the xaxis.
(This only works for real solutions).
In the picture below, the left parabola has 2 real solutions (red dots), the middle parabola has 1 existent solution (red dot) and the right nigh parabola has no real solutions (yes, information technology does have imaginary ones).
What does the discriminant look like?
Answer
It looks like .. a number.
5, 2, 0, one
– each of these numbers is the discriminant for iv different quadratic equations.
What is the discriminant anyway?
Answer
The discriminant is a
number
that can be calculated from any quadratic equation.
A quadratic equation is an equation that can be written equally $$ ax^ii + bx + c $$
(where $$a \ne 0 $$).
What is the formula for the Discriminant?
Answer
The discriminant for any quadratic equation of the form
$$ y =\red a 10^ii + \blue bx + \color {green} c $$
is found past the following formula and it provides critical information regarding the nature of the roots/solutions of whatever quadratic equation.
$ \boxed{Formula} \\ \text{Discriminant } = \blue b^two iv \red a \color{green} c $
$ \boxed{Example} \\ \text{Equation :} y =\red 3 10^2 + \blueish 9x + \colour {lightgreen} 5 \\ \text{Discriminant } = \blue ix^2 4 \cdot \red 3 \cdot \color{green} 5 \\ \text{Discriminant } = \boxed{ six} $
What does this formula tell usa?
Respond
The discriminant tells us the following information nigh a quadratic equation:
 If the solution is a real number or an imaginary number.
 If the solution is rational or if it is irrational.
 If the solution is one unique number or 2 different numbers.
Nature of the
Solutions
Value of the discriminant  Blazon and number of Solutions  Example of graph 

$ b^2 – 4ac > 0 $ $ \text{Example :} \\ y = \carmine 3x^ii \blueish{six}x + \color{green} ii \\ \text{Discriminant} \\ \blue{6}^ii – 4 \cdot \cherryred 3 \cdot \color{green} 2 \\ = \boxed{12} \\ $ 
If the discriminant is a positive number, then there are 2 real solutions. This means that the graph of the parabola interepts the xaxis at 2 singledout points . 

$ \text{Example :} \\ y = \carmine 3x^2 + \bluish 4 x \color{green} {4} \\ \text{Discriminant} \\ \blue four^2 – 4 \cdot \crimson 3 \cdot \colour{green} {4} \\ = \boxed{64} \\ $ 
If the discriminant is positive and also a perfect square like 64, then there are 2 real 

$ \text{Example :} \\ y = \rubyred 3x^2 \blue {6} x + \colour{green} 2 \\ \text{Discriminant} \\ \blue {6}^2 – iv \cdot \ruby iii \cdot \color{green} ii \\ = \boxed{12 } \\ $ 
If the discriminant is positive and 
$ b^2 – 4ac = 0 $ $ \text{Example :} \\ y = \red 4 x^2 \blueish{28}x + \colour{lightgreen} {49} \\ \text{Discriminant} \\ \blue{28}^ii – iv \cdot \red 4 \cdot \colour{green} 49 \\ = \boxed{0} \\ $ 

$ b^2 – 4ac < 0 $ $ \text{Example :} \\ y = \red 10^2 \blue{three}x + \color{lightgreen} 4 \\ \text{Discriminant} \\ \blue{iii}^2 – iv \cdot \red i \cdot \color{green} iv \\ = \boxed{7} \\ $ 
There are only imaginary Solutions. This means that the graph of the quadratic never intersects the axes. 
Case
Quadratic Equation:
$$ y = x^2 + 2x + one$$
$ a = \rubyred one \\ b = \blue 2 \\ a = \color{green} {1} $
The discriminant for this equation is:
$ \text{Discriminant } = \blue b^2 4 \red a \color{darkgreen} c \\ \text{Discriminant } = \blue 2^2 four \cdot \red 1 \cdot \color{green} \cdot 1 \\ \text{Discriminant } = 4 4 \\ \text{Discriminant } = \boxed{0} \\ $
Since the discriminant is zero, at that place should be 1 real solution to this equation.
Below is a picture representing the graph and the one solution of
$$ y = x^2 + 2x + 1$$.
Practice
Problems
Practice i
In this quadratic equation,
$$ y =\bloodred 1 x^ii + \blue {2} + \colour {green} 1 $$
$ \text{Equation : } y =\cerise i x^2 + \blueish {2}10 + \colour {green} 1 \\ a = \scarlet 1 \\ b = \blueish{ii} \\ c = \color{green} 1 $
Using our general formula:
$$ \text{Discriminant } \\ \begin{aligned} &= \blue b^2 4 \cdot \red a \cdot \color{green} c \\ &= \blue {two}^2 4 \cdot \carmine 1 \cdot \colour{green} 1 \\ &= \boxed{0} \terminate{aligned} $$
Since the discriminant is aught, we should look 1 existent solution which you can see pictured in the graph beneath.
Practice two
In this quadratic equation,
$$ y =\red 1 x^2 + \blue {1}10 + \color {green} 1 $$
$ \text{Equation : } y =\red 1 ten^2 + \blueish {one} + \colour {green} 1 \\ a = \ruddy ane \\ b = \blue{1} \\ c = \color{greenish} {2} $
Using our general formula:
$$ \text{Discriminant } \\ \brainstorm{aligned} &= \blue b^2 four \cdot \red a \cdot \color{green} c \\ &= \blue {i}^2 4 \cdot \red 1 \cdot \color{darkgreen} {two} \\ &= i – 8 \\ &= 1 + 8 = \boxed 9 \end{aligned} $$
Since the discriminant is positive and rational, in that location should be 2 real rational solutions to this equation. Every bit you lot can see beneath, if you apply the quadratic formula to notice the actual solutions, you do indeed get two real rational solutions.
Do 3
In this quadratic equation,
y =
one10²
− 1
.
 a = i
 b = 0
 c = − 1
$$ \colour{Crimson}{b^two} – 4\colour{Magenta}{a}\color{Blue}{c} \\ \color{Rubyred}{(0)^two} – four\color{Magenta}{(1)}\colour{Blue}{(1)} = iv $$
Since the discriminant is positive and a perfect foursquare, we have two real solutions that are rational.
Again if yous’d like to come across the bodily solutions and the graph, merely wait below:
Practise four
In this quadratic equation,
y = x² + 4x − 5.
 a = 1
 b = 4
 c = − v
$$ \color{Red}{b^2} – 4\colour{Magenta}{a}\color{Blue}{c} \\ \color{Red}{(4)^2} – 4\colour{Magenta}{(one)}\color{Blue}{(5)} \\ xvi – iv(5) = 16 +twenty \\ = 36 $$
Since this quadratic equation’s discriminant is positive and a perfect square, there are two existent solutions that are rational.
Practise 5
In this quadratic equation,
y = ten² – 4x + 5.
 a = one
 b = 4
 c = v
$$ \colour{Red}{b^2} – 4\color{Magenta}{a}\colour{Bluish}{c} \\ \color{Ruddy}{(4)^2} – 4\color{Magenta}{(1)}\color{Blueish}{(5)} \\ = 16 – 20 = 4 $$
Since the discriminant is negative, there are no real solutions to this quadratic equation. The simply solutions are imaginary.
Below is a picture of this quadratic’s graph.
Practice half dozen
y = x² + 4
 a = one
 b = 0
 c = 4
$$ \color{Red}{b^2} – 4\color{Magenta}{a}\color{Blue}{c} \\ \color{Scarlet}{(0)^two} – 4\colour{Magenta}{(i)}\color{Blue}{(four)} = sixteen $$
Since the discriminant is negative, in that location are two imaginary solutions to this quadratic equation.
The solutions are
2i
and
2i.
Below is a picture of this equations graph.
Practice 7
y = x² + 25
 a = ane
 b = 0
 c = 25
$$ \color{Red}{b^2} – 4\colour{Magenta}{a}\color{Bluish}{c} \\ \color{Crimson}{(0)^two} – iv\color{Magenta}{(1)}\color{Blue}{(25)} = 100 $$
Since the discriminant is negative, there are ii imaginary solutions to this quadratic equation.
The solutions are
5i
and
5i.
Which Function Has a Negative Discriminant Value
Source: https://www.mathwarehouse.com/quadratic/discriminantinquadraticequation.php