How many real solutions does the equation have calculator

Discriminant Calculator

Table of Contents Show

• How to Use the Discriminant Calculator?
• Discriminant Definition
• Standard Form
• Frequently Asked Questions on Discriminant Calculator
• Why is discriminant value important in quadratic equation?
• How to determine the nature of roots using the discriminant value?
• Is it Quadratic?
• How Does this Work?
• What is the discriminant?
• How many real solutions can a quadratic equation have?
• Why is the number of solutions useful?
• How to find the number of solutions to a quadratic equation
• Study summary
• Do it yourself!
• Extra credit
• How many real solutions does have?
• How many real solutions a quadratic equation has?
• Which equation has real solution?

Enter values of a,b,c in ax2+bx+c=0

The discriminant calculator is a free online tool that gives the discriminant value for the given coefficients of a quadratic equation. BYJU’S online discriminant calculator tool makes the calculations faster and easier, where it displays the value in a fraction of seconds.

How to Use the Discriminant Calculator?

The procedure to use the discriminant calculator is as follows:

Step 1: Enter the coefficient values such as “a”, “b” and “c” in the given input fields

Step 2: Now click the button “Solve” to get the output

Step 3: The discriminant value will be displayed in the output field

Discriminant Definition

A discriminant is a function of the coefficients of a polynomial equation that expresses the nature of the roots of the given quadratic equation. The equations can discriminate between the possible types of answer, such as:

• When the discriminant value is positive, we get two real solutions
• When the discriminant value is zero, we get one real solution
• When the discriminant value is negative, we get a pair of complex solutions

Standard Form

The standard discriminant form for the quadratic equation ax2 + bx + c = 0 is

Discriminant, D = b2 – 4ac

Where

a is the coefficient of x2

b is the coefficient of x

c is a constant term.

Frequently Asked Questions on Discriminant Calculator

Why is discriminant value important in quadratic equation?

Discriminant value reveals the nature of the roots of the quadratic equation. The roots of the quadratic equation may be either real or complex. It helps to determine the solution of an equation.

How to determine the nature of roots using the discriminant value?

A quadratic equation can have different types of roots. Based on the discriminant value, the nature of roots are determined. They are:

If D> 0, the roots are real and unequal

If D = 0, the roots are real and equal

If D < 0, the roots are not real (i.e., complex)

We can help you solve an equation of the form "ax2 + bx + c = 0"
Just enter the values of a, b and c below:

algebra/images/quadratic-solver.js

Is it Quadratic?

Only if it can be put in the form ax2 + bx + c = 0, and a is not zero.

The name comes from "quad" meaning square, as the variable is squared (in other words x2).

These are all quadratic equations in disguise:

In disguise	In standard form	a, b and c
x2 = 3x -1	x2 - 3x + 1 = 0	a=1, b=-3, c=1
2(x2 - 2x) = 5	2x2 - 4x - 5 = 0	a=2, b=-4, c=-5
x(x-1) = 3	x2 - x - 3 = 0	a=1, b=-1, c=-3
5 + 1/x - 1/x2 = 0	5x2 + x - 1 = 0	a=5, b=1, c=-1

How Does this Work?

The solution(s) to a quadratic equation can be calculated using the Quadratic Formula:

The "±" means we need to do a plus AND a minus, so there are normally TWO solutions !

The blue part (b2 - 4ac) is called the "discriminant", because it can "discriminate" between the possible types of answer:

• when it is positive, we get two real solutions,
• when it is zero we get just ONE solution,
• when it is negative we get complex solutions.

Learn more at Quadratic Equations

Note: you can still access the old version here.

Explore Quadratic equations

Picture it: You’re at the kitchen table, with a convoluted quadratic equation in front of you. You don’t want to solve it. You don’t even need to solve it. All you actually need to find is the number of solutions.

Is that even possible?

You’ll be happy to know that YES, it is!

That’s right — we can find just the number of solutions without having to find the solutions themselves! (cue happy dance)

But how do we find the number of solutions?

We use the discriminant!

What is the discriminant?

The discriminant of a quadratic equation written in standard form $$ax^{2}+bx+c=0$$ is the value of the expression:

$$b^{2}-4ac$$

This is the value that determines the number of solutions to a quadratic equation — so basically, it’s our new best friend.

How many real solutions can a quadratic equation have?

Quadratic equations can have $$0$$, $$1$$, or $$2$$ solutions.

The number of real solutions of a quadratic equation depends on the sign of the discriminant $$b^2-4ac$$ of that quadratic equation.

Why is the number of solutions useful?

Quite frankly, we don’t always need the exact values of solutions; sometimes, we just need to know how many there are.

For example, we may want to know if the related graph intersects the $$x$$-axis and, if it does, at how many points. We get that information from the number of solutions of a quadratic equation!

How to find the number of solutions to a quadratic equation

Okay, so you’ve found yourself in a situation where you only need to know the number of solutions — but you’re not sure where to start.

Luckily, we do! Let’s walk through an example together.

Example

Our goal is to find the number of solutions to this quadratic equation:

$$9x^{2} + 42x + 49 = 0$$

Our first order of business is to identify the coefficients $$a$$, $$b$$, and $$c$$ (this helps us calculate the discriminant):

$$a=9, b=42, c=49$$

We’ve identified the coefficients, so we can evaluate the discriminant by substituting these coefficients into the expression $$b^2-4ac$$:

$$42^2-4\times9\times49$$

Now, we’ll evaluate the power (remember your PEMDAS!):

$$1764-4\times9\times49$$

Next is calculating the product:

$$1764-1764$$

The sum of two opposites equals to $$0$$, so:

$$0$$

The discriminant equals $$0$$, which means the quadratic equation has one real solution!

$$1\text{ real solution}$$

Not so bad once you see it in context!

Let’s summarize the steps so you can apply it beyond this example:

Study summary

1. Rewrite the quadratic equation in standard form (if it’s not already).
2. Identify the coefficients.
3. Calculate the discriminant of the quadratic equation.
4. Based on the sign of the discriminant, determine the number of solutions of the corresponding quadratic equation.

Do it yourself!

If you’ve been around here awhile, you know we’re big advocates of putting your learnings into practice. Once you get the hang of it, it won’t take long to make these calculations — but step one is to truly get the hang of it! That’s where these problems come in.

Find the number of solutions to the following quadratic equations:

1. $$(x-1)(x+1)=x+1$$
2. $$x^2+x=0$$
3. $$-2y^{2} = 0$$
4. $$x^2+2x=-\frac{5}{2}$$

Solutions:

1. $$2$$ real solutions
2. $$2$$ real solutions
3. $$1$$ real solution
4. No real solutions

How did that practice feel? Do you need some additional guidance?

Try scanning a practice problem (or any other problem!) with your Photomath app so we can walk through each step.

Here’s a sneak peek of what you’ll see:

Extra credit

How to visualize the relation of the discriminant and the number of solutions:

Want to go a step deeper? If you’re a visual learner, this will help you understand why the discriminant tells us the number of solutions!

The graph of a quadratic function is a parabola, or a sort of U-shaped curve. When placed in an $$xy$$- coordinate system, it faces upwards or downwards.

When we look for the number of solutions to a quadratic equation written in standard form $$ax^2+bx+c=0$$, we’re also looking for the number of $$x$$-intercepts of the related parabola. That parabola can intersect the $$x$$-axis in zero, one or two points:

• Related quadratic equation has no real solutions, meaning that the discriminant is less than 0

• Related quadratic equation has one real solution, meaning that the discriminant equals 0

• 

  Related quadratic equation has two real solutions, meaning that the discriminant is greater than 0

Take any quadratic equation you want! If you follow these steps, you’ll notice the same things:

1. Solve it and count the number of solutions.
2. Compare the value of the discriminant to $$0$$.
3. Use your graphic calculator to plug in the related quadratic function and look for the number of $$x$$-intercepts.

Pretty cool when it clicks, right?

How many real solutions does have?

How many real solutions a quadratic equation has?

Which equation has real solution?