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### Course: Algebra (all content) > Unit 9

Lesson 7: The quadratic formula- The quadratic formula
- Understanding the quadratic formula
- Using the quadratic formula
- Worked example: quadratic formula
- Worked example: quadratic formula (example 2)
- Worked example: quadratic formula (negative coefficients)
- Quadratic formula
- Using the quadratic formula: number of solutions
- Number of solutions of quadratic equations
- Proof of the quadratic formula
- Quadratic formula review
- Discriminant review
- Quadratic formula proof review

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# Discriminant review

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

for any quadratic equation like:

## What is the discriminant?

The ${\text{discriminant}}$ is the part of the quadratic formula under the square root.

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:

From the equation, we see:

$a=6$ $b=10$ $c=-1$

Plugging these values into the discriminant, we get:

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

This makes sense if we think about the corresponding graph of $y=6{x}^{2}+10x-1$ :

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: $6{x}^{2}+10x-1=0$ .

## Want to join the conversation?

- Why do we need the discriminant? We already know what kind of solutions there are when we solve using the quadratic formula.(7 votes)
- 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.(98 votes)

- "A discriminant of zero indicates that the quadratic has a repeated real number solution." what exactly does this mean?(19 votes)
- It means that you only have one solution(29 votes)

- I don't understand what F(x) means? The f symbol just appeared(0 votes)
- 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.(40 votes)

- how can the discriminant help graph?(3 votes)
- 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(16 votes)

- How do you find the discriminant from looking at a graph?(3 votes)
- 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.(16 votes)

- 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.(4 votes)- 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.(1 vote)

- how discriminant decides what are the nature of the two roots?

I mean how?(1 vote)- The quadratic formula: x = [-B +/- sqrt(B^2-4AC)] / (2A)

The discriminant is B^2-4AC. Notice this is the portion of the formula inside the square root.

If the discriminant = 0, then the formula degrades to x = -B/(2A). So, there is only one solution.

If the discriminant is positive, then the square root creates a real number. So, there are 2 real solutions.

If the discriminant is negative, then the square root is not a real number. Square roots of negative values require the using of complex numbers. So, there are 2 solutions, that are not real numbers. Or, 2 complex solutions.

Hope this helps.(16 votes)

- How is a quadratic equation with a negative discriminant graphed?(2 votes)
- 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.(12 votes)

- 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?(3 votes)- 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.(7 votes)

- I dont understand the graph!(4 votes)
- The graph is a y=x^2 graph. The formula has an "f" to help with the quadratic formula. You should ask a more specific question.(4 votes)