"A discriminant of zero indicates that the quadratic has a repeated real number solution." what exactly does this mean?

It means that you only have one solution

Why do we need the discriminant? We already know what kind of solutions there are when we solve using the quadratic formula.

𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 ⇒ 𝑥 = (-𝑏 ± √(𝑏² – 4𝑎𝑐))/(2𝑎) Using this formula, it is advisable to calculate the discriminant, 𝑏² – 4𝑎𝑐, first because if it is negative we know that there are no real solutions and we can skip the rest of the calculations.

how can the discriminant help graph?

It determines the number of times the graph crosses the x-axis. Discriminant > 0: the graph crosses the x-axis twice Discriminant = 0: the graph touches the x-axis at its maximum or minimum point Discriminant < 0: The graph has no x-intercepts, which means it is wholly above or below the x-axis

I have a question that was given to me in class, it is: x^2 - (k+4)x + k + 7 = 0. Find k. Answer: k = -6 & 2 I understand HOW to put this into the discriminant and get the correct answer, but not WHY we do that. How come we have to use the discriminant to find k? How do I know when I need to use this for equations?

The answers that you found (for k) are when the discriminant equal 0 (b^2-4ac=0) -- which means that the function has only one solution. *When you graph* (k+4)^2-4(k+7), you get a convex parabola with vertex (-2,-16) and x-intercepts at (-6,0) and (2,0). *That implies* that for k; -6<k<2, that the discriminant is negative. In other words there is no real solution for those values of k. For k=-6 & k=2, which you found the function (with x) has only one x-intercept (which is the vertex). For k<-6 & k>2, the function has two solutions (x-intercepts). *So*, you find the discriminant in order to figure out for which values for k, the function has 0, 1 or 2 solutions.

How do you find the discriminant from looking at a graph?

I don't think there's an easy way to find the exact value of the discriminant by looking at the graph, but looking at the graph can tell you if the discriminant is positive, negative, or zero. If the graph doesn't touch the x axis at all, the discriminant is negative If the graph touches the x axis a only one point, the discriminant is zero If the graph touches the x axis at two distinct points, the discriminant is positive. Sorry I couldn't give you an easy answer, but if you know the equation, then it's pretty easy to find the discriminant, so I don't know if it's worth it to learn how to find it from only the graph.

How is a quadratic equation with a negative discriminant graphed?

You just don't have x-intercepts to work with. You can graph it using a table of values -- pick values for X and calculate Y for each X. You can still find the vertex and axis of symmetry.

I don't understand what F(x) means? The f symbol just appeared

f(x) is read as f of x, and it means a function in terms of x. This is called functional notation, and it has the same meaning as y = at this point in math. As you get into Algebra II, you will learn how to combine functions where this language will be more useful than the y = form of equations. The biggest use of f(x) in Algebra I is when you are asked to find a specific value of x. So if f(x) = 2x + 6, this is equivalent to y = 2x+ 6, but if I wanted to find the value of the function at x = 8, with functional notation, I could just say f(8) which is solved by putting 8 into x and getting f(8) = 22.

Main content

Algebra 1

Course: Algebra 1 > Unit 14

Lesson 6: The quadratic formula

Discriminant review

Q: I have a question that was given to me in class, it is: x^2 - (k+4)x + k + 7 = 0. Find k. Answer: k = -6 & 2 I understand HOW to put this into the discriminant and get the correct answer, but not WHY we do that. How come we have to use the discriminant to find k? How do I know when I need to use this for equations?

The answers that you found (for k) are when the discriminant equal 0 (b^2-4ac=0) -- which means that the function has only one solution. *When you graph* (k+4)^2-4(k+7), you get a convex parabola with vertex (-2,-16) and x-intercepts at (-6,0) and (2,0). *That implies* that for k; -6<k<2, that the discriminant is negative. In other words there is no real solution for those values of k. For k=-6 & k=2, which you found the function (with x) has only one x-intercept (which is the vertex). For k<-6 & k>2, the function has two solutions (x-intercepts). *So*, you find the discriminant in order to figure out for which values for k, the function has 0, 1 or 2 solutions.

CCSS.Math:

HSA.REI.B.4

HSA.REI.B.4b

Google Classroom

The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.

Quick review of the quadratic formula

The quadratic formula says that

x, equals, start fraction, minus, start color #e07d10, b, end color #e07d10, plus minus, square root of, start color #e07d10, b, end color #e07d10, squared, minus, 4, start color #7854ab, a, end color #7854ab, start color #e84d39, c, end color #e84d39, end square root, divided by, 2, start color #7854ab, a, end color #7854ab, end fraction

for any quadratic equation like:

start color #7854ab, a, end color #7854ab, x, squared, plus, start color #e07d10, b, end color #e07d10, x, plus, start color #e84d39, c, end color #e84d39, equals, 0

What is the discriminant?

The start color #e07d10, start text, d, i, s, c, r, i, m, i, n, a, n, t, end text, end color #e07d10 is the part of the quadratic formula under the square root.

x, equals, start fraction, minus, b, plus minus, square root of, start color #e07d10, b, squared, minus, 4, a, c, end color #e07d10, end square root, divided by, 2, a, end fraction

The discriminant can be positive, zero, or negative, and this determines how many solutions there are to the given quadratic equation.

A positive discriminant indicates that the quadratic has two distinct real number solutions.
A discriminant of zero indicates that the quadratic has a repeated real number solution.
A negative discriminant indicates that neither of the solutions are real numbers.

Want to understand these rules at a deeper level? Check out this video.

Example

We're given a quadratic equation and asked how many solutions it has:

6, x, squared, plus, 10, x, minus, 1, equals, 0

From the equation, we see:

a, equals, 6
b, equals, 10
c, equals, minus, 1

Plugging these values into the discriminant, we get:

\begin{aligned} &b^2-4ac\\\\ =&10^2-4(6)(-1)\\\\ =&100+24\\\\ =&124 \end{aligned}

This is a positive number, so the quadratic has two solutions.

This makes sense if we think about the corresponding graph.

Notice how it crosses the x-axis at two points. In other words, there are two solutions that have a y-value of 0, so there must be two solutions to our original equation: 6, x, squared, plus, 10, x, minus, 1, equals, 0.

Practice

Problem 1

Current

f, left parenthesis, x, right parenthesis, equals, 3, x, squared, plus, 24, x, plus, 48

What is the value of the discriminant of f?

How many distinct real number zeros does f have?

Want more practice? Check out this exercise.

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drossington
Posted 6 years ago. Direct link to drossington's post “Why do we need the discri...”
Why do we need the discriminant? We already know what kind of solutions there are when we solve using the quadratic formula.
Button navigates to signup pageComment on drossington's post “Why do we need the discri...”
(7 votes)
Answer
- Jerry Nilsson
  Posted 6 years ago. Direct link to Jerry Nilsson's post “𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 ⇒ �...”
  𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 ⇒ 𝑥 = (-𝑏 ± √(𝑏² – 4𝑎𝑐))/(2𝑎)
  
  Using this formula, it is advisable to calculate the discriminant, 𝑏² – 4𝑎𝑐, first because if it is negative we know that there are no real solutions and we can skip the rest of the calculations.
  Comment on Jerry Nilsson's post “𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 ⇒ �...”
  (79 votes)
Shuss824
Posted 6 years ago. Direct link to Shuss824's post “"A discriminant of zero i...”
"A discriminant of zero indicates that the quadratic has a repeated real number solution." what exactly does this mean?
Button navigates to signup pageComment on Shuss824's post “"A discriminant of zero i...”
(17 votes)
Answer
- FightingJ
  Posted 5 years ago. Direct link to FightingJ's post “It means that you only ha...”
  It means that you only have one solution
  Comment on FightingJ's post “It means that you only ha...”
  (24 votes)
Kathy Downey
Posted 6 years ago. Direct link to Kathy Downey's post “I don't understand what F...”
I don't understand what F(x) means? The f symbol just appeared
Button navigates to signup pageComment on Kathy Downey's post “I don't understand what F...”
(1 vote)
Answer
- David Severin
  Posted 6 years ago. Direct link to David Severin's post “f(x) is read as f of x, a...”
  f(x) is read as f of x, and it means a function in terms of x. This is called functional notation, and it has the same meaning as y = at this point in math. As you get into Algebra II, you will learn how to combine functions where this language will be more useful than the y = form of equations. The biggest use of f(x) in Algebra I is when you are asked to find a specific value of x. So if f(x) = 2x + 6, this is equivalent to y = 2x+ 6, but if I wanted to find the value of the function at x = 8, with functional notation, I could just say f(8) which is solved by putting 8 into x and getting f(8) = 22.
  Comment on David Severin's post “f(x) is read as f of x, a...”
  (35 votes)
Anirudh Parmar
Posted 5 years ago. Direct link to Anirudh Parmar's post “if the eqaution has no re...”
if the eqaution has no real roots , use the discriminant to determine the value of n.
0=5.5x^2+nx+n and the discriminant is -40.

This is another homework question I dont know how to do this.
Button navigates to signup pageComment on Anirudh Parmar's post “if the eqaution has no re...”
(5 votes)
Answer
- rylan.wetsell
  Posted 6 months ago. Direct link to rylan.wetsell's post “basically you're looking ...”
  basically you're looking b and c, which in this case are the same, so you can plug everything into the discriminant equation (b^2 -4ac):
  n^2 -4(5.5)(n)=-40
  i don't know if i'm being dumb and there's an easier way to solve this but you can simplify this to:
  n^2 -11n +40 =0
  which, you'll notice, is a quadratic equation, so you just solve for that to get n.
  Button navigates to signup page
  (1 vote)
westina_7
Posted 4 years ago. Direct link to westina_7's post “how can the discriminant ...”
how can the discriminant help graph?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Isabella C
  Posted 4 years ago. Direct link to Isabella C's post “It determines the number ...”
  It determines the number of times the graph crosses the x-axis.
  Discriminant > 0: the graph crosses the x-axis twice
  Discriminant = 0: the graph touches the x-axis at its maximum or minimum point
  Discriminant < 0: The graph has no x-intercepts, which means it is wholly above or below the x-axis
  Button navigates to signup page
  (15 votes)
Sage
Posted 3 years ago. Direct link to Sage's post “How do you find the discr...”
How do you find the discriminant from looking at a graph?
Button navigates to signup pageComment on Sage's post “How do you find the discr...”
(2 votes)
Answer
- Hannah Alisse
  Posted 3 years ago. Direct link to Hannah Alisse's post “I don't think there's an ...”
  I don't think there's an easy way to find the exact value of the discriminant by looking at the graph, but looking at the graph can tell you if the discriminant is positive, negative, or zero.
  
  If the graph doesn't touch the x axis at all, the discriminant is negative
  If the graph touches the x axis a only one point, the discriminant is zero
  If the graph touches the x axis at two distinct points, the discriminant is positive.
  
  Sorry I couldn't give you an easy answer, but if you know the equation, then it's pretty easy to find the discriminant, so I don't know if it's worth it to learn how to find it from only the graph.
  Comment on Hannah Alisse's post “I don't think there's an ...”
  (13 votes)
sunix777
Posted 6 years ago. Direct link to sunix777's post “How is a quadratic equati...”
How is a quadratic equation with a negative discriminant graphed?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “You just don't have x-int...”
  You just don't have x-intercepts to work with.
  You can graph it using a table of values -- pick values for X and calculate Y for each X.
  You can still find the vertex and axis of symmetry.
  Comment on Kim Seidel's post “You just don't have x-int...”
  (11 votes)
Ethan
Posted 20 days ago. Direct link to Ethan's post “My name is ethan”
My name is ethan
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(7 votes)
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bondwad
Posted 17 days ago. Direct link to bondwad's post “my name is ethan”
my name is ethan
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(4 votes)
Answer
- loperob
  Posted 17 days ago. Direct link to loperob's post “my name is ethan”
  my name is ethan
  Button navigates to signup page
  (4 votes)
Bree
Posted 3 years ago. Direct link to Bree's post “I have a question that wa...”
I have a question that was given to me in class, it is:
x^2 - (k+4)x + k + 7 = 0. Find k.
Answer: k = -6 & 2

I understand HOW to put this into the discriminant and get the correct answer, but not WHY we do that. How come we have to use the discriminant to find k? How do I know when I need to use this for equations?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Timo
  Posted 3 years ago. Direct link to Timo's post “The answers that you foun...”
  The answers that you found (for k) are when the discriminant equal 0 (b^2-4ac=0) -- which means that the function has only one solution.
  When you graph (k+4)^2-4(k+7), you get a convex parabola with vertex (-2,-16) and x-intercepts at (-6,0) and (2,0).
  That implies that for k; -6<k<2, that the discriminant is negative. In other words there is no real solution for those values of k.
  For k=-6 & k=2, which you found the function (with x) has only one x-intercept (which is the vertex).
  For k<-6 & k>2, the function has two solutions (x-intercepts).
  So, you find the discriminant in order to figure out for which values for k, the function has 0, 1 or 2 solutions.
  Comment on Timo's post “The answers that you foun...”
  (6 votes)