How to know how many real solutions an equation has

When dealing with quadratic equations, we often see either two real solutions or two complex solutions.  However, it is also possible that a quadratic will have exactly one solution.

So, when does a quadratic have one solution?  A quadratic equation has one solution when the discriminant is zero.  From an algebra standpoint, this means b2 = 4ac.  Visually, this means the graph of the quadratic (a parabola) will have its vertex resting on the x-axis.

Of course, we can also look at the coefficients in a quadratic equation to tell if it has one solution.

In this article, we’ll talk about how you can tell that a quadratic has one solution.  We’ll also look at some examples, along with how to write a quadratic equation given that it only has one solution.

Let’s get started.

There are a few different ways to tell when a quadratic equation has one solution:

• Look at the discriminant – if it is zero, there is only one solution to the quadratic.
• Look at the graph – if the vertex of the parabola rests on the x-axis, there is only one solution to the quadratic.
• Look at the coefficients – there is a special pattern that will tell you when there is only one solution to the quadratic (more on this later in the article!)

We’ll start with a method that uses the discriminant.

Look At The Discriminant

The first way to tell if a quadratic has one solution is to look at the discriminant.  If the discriminant is zero, then the quadratic equation has only one real solution.

Remember that for the quadratic equation given by

• ax2 + bx + c = 0

the discriminant is the expression b2 – 4ac.  The discriminant is the expression under the radical sign in the quadratic formula:

To get a discriminant of zero, we need to set b2 – 4ac equal to zero.  This gives us:

• b2 – 4ac = 0
• b2 = 4ac

Here is one example of a quadratic equation with only one solution:

•  x2 – 6x  + 9

In this case, a = 1, b = -6, and c = 9.  This gives us:

• b2 = (-6)2 = 36
• 4ac = 4(1)(9) = 36

So, b2 = 4ac, and thus the discriminant is zero.  This means that the quadratic has only one solution: x = 3.

You can find this solution by using the quadratic formula.  You can also find it by factoring the quadratic:

• x2 – 6x  + 9 = (x – 3)(x – 3)

Look At The Graph

Another way to tell if a quadratic has one solution is to look at the graph of the quadratic.  For any quadratic equation, its graph will be a parabola.

Remember that a key feature of a parabola is its vertex.  The vertex of a parabola is sort of like the “mountain top” (for negative values of a) or “valley bottom” (for positive values of a).

As you can see in the graph pictured above, the vertex (valley bottom) of this parabola lies on the x axis. This means the quadratic equation x2 – 4x + 4 has one real solution (at x = 2).

Be careful: for a quadratic equation to have only one real solution (a double root), its graph must touch the x axis exactly once.

If the graph touches the x axis twice, then it has two distinct real solutions.

If the graph does not touch the x axis at all, then it has two complex solutions (and no real solutions).

Look At The Coefficients

You can also look at the coefficients of a quadratic equation in standard form to tell if it has one real solution.  Remember that the standard form of a quadratic equation has zero on one side, and terms in descending order on the other:

• ax2 + bx + c = 0

There are two possible cases to deal with here: either a = 1, or a is not equal to 1.

When The Coefficient of x2 Is Equal To 1 (a = 1)

In this case, we look for c = b2 / 4.  If this is true, then the quadratic has one solution, which is x = -b / 2.

For example, the quadratic equation x2 + 8x + 16 has one solution.  In this case, a = 1, b = 8, and c = 16. Then we have:

• b2 / 4
• = 82 / 4
• = 64 / 4
• = 16

which is the same as the value of c.  The one solution is:P

• x = -b / 2
• x = -8/2
• x = -4

We can also find this solution by factoring the quadratic as:

• x2 + 8x + 16 = (x + 4)(x + 4)

which gives us a solution of x = -4 (a real repeated root). We can also use the quadratic formula with a = 1, b = 8, and c = 16 to get the same result of x = -4.

When The Coefficient of x2 Is Not Equal To 1 (a is not equal to 1)

In this case, divide the entire quadratic equation by a.  Then, you are in the first case, when the coefficient of x2 is equal to 1.

For example, consider the equation:

• 2x2 + 12x + 18 = 0

Since a = 2, we will divide both sides by 2.

This leaves us with:

• x2 + 6x + 9 = 0

For this new quadratic equation, we have a = 1, b = 6, and c = 9.

Then we have:

• b2 / 4
• = 62 / 4
• = 36 / 4
• = 9

which is the same as the value of c.  The one solution is x = -b / 2 = -6 / 2 = -3.

We can also find this solution by factoring the quadratic as:

• x2 + 6x + 9 = (x + 3)(x + 3)

which gives us a solution of x = -3 (a real repeated root). We can also use the quadratic formula with a = 1, b = 6, and c = 9 to get the same result of x = -3.

Remember that you can always use a calculator to help you verify the solutions to a quadratic equation.  You can also use a quadratic equation solver, such as this one from WolframAlpha.

For WolframAlpha’s calculator, remember that:

• The quadratic coefficient (x2 coefficient) means a
• The linear coefficient (x coefficient) means b
• The constant coefficient means c

Examples of Quadratic Equations With One Solution

Here are some examples of quadratic equations with one solution.  Look at them to see if you notice a pattern before reading further.

• x2 + 2x + 1 = 0 (solution: x = -1)
• x2 + 4x + 4 = 0 (solution: x = -2)
• x2 + 6x + 9 = 0 (solution: x = -3)
• x2 + 8x + 16 = 0 (solution: x = -4)
• x2 + 10x + 25 = 0 (solution: x = -5)

One thing you might notice is that the x coefficients (b values) are all even numbers:

• 2, 4, 6, 8, and 10

These come from doubling the numbers 1, 2, 3, 4, and 5.

Another thing you might notice is that the constant terms (c values) are all perfect squares:

• 1, 4, 9, 16, and 25

These come from squaring the numbers 1, 2, 3, 4, and 5.

One last thing you might notice is that the solutions are the negatives of the numbers 1, 2, 3, 4, and 5.

After we notice the pattern, we see the beauty of it: you can pick any whole number n and create a quadratic with a single solution (the solution is n).

Here’s how to do it.

• First, choose your whole number n. Remember that a whole number has no fractions or decimals.  For this example, I choose n = 7.
• Next, calculate 2n.  This will be the value of b.  For this example, b = 2n = 2*7 = 14.
• Then, calculate n2.  This will be the value of c.  For this example, c = n2 = 72 = 49.
• Finally, your quadratic with one solution has a = 1, b = 2n, and c = n2.  So, the quadratic would look like x2 + 2n + n2.  The solution will be –n (the negative of the whole number n you chose at the beginning).

For this example, our quadratic with one solution is x2 + 14x + 49.  The solution is x = -7.

We can also change the sign of all the b values to get an entire new set of quadratic equations with only one solution:

• x2 – 2x + 1 = 0 (solution: x = 1)
• x2 – 4x + 4 = 0 (solution: x = 2)
• x2 – 6x + 9 = 0 (solution: x = 3)
• x2 – 8x + 16 = 0 (solution: x = 4)
• x2 – 10x + 25 = 0 (solution: x = 5)

Finally, we can take any given quadratic equation with one solution and multiply by any number (except zero) to get a new quadratic equation with the same solution.

How Do You Write A Quadratic Equation With One Solution?

Now it’s time to think about working backwards.  That means taking one number and writing a quadratic equation whose only solution is that number.

This is easy to do.  Even better, there are infinitely many such equations for a given solution.

Let’s say we want to write a quadratic equation with only one solution.  Let’s also assume that we want the solution to be the number n.

To do this, we would simply write the equation:

• (x – n)(x – n) = 0

After using FOIL on the parentheses, we would get the equation

• x2 – 2nx + n2 = 0

This equation has exactly one solution, which is x = n (it is a double real root).

Example: Writing A Quadratic Equation With One Real Solution

For example, let’s say I want a quadratic equation with a solution of n = 3.

Then the equation would be

• (x – 3)(x – 3) = 0

After using FOIL, we would get the equation

• x2 – 6x + 9 = 0

This equation has exactly one solution, which is the value x = 3.  We can also generate as many more equations as we like by simply multiplying both sides by 2, 3, 4, 5, and so forth:

• 2x2 – 12x + 18 = 0 (multiplied by 2)
• 3x2 – 18x + 27 = 0 (multiplied by 3)
• 4x2 – 24x + 36 = 0 (multiplied by 4)
• 5x2 – 30x + 45 = 0 (multiplied by 5)

Does A Quadratic Equation Always Have Two Solutions?

In terms of real solutions, there are always either 0, 1, or 2 real solutions to a quadratic equation, depending on the sign of the discriminant.

However, there is always at least 1 solution if you count both real and complex numbers.

• If the discriminant is positive, there are exactly 2 real solutions (they are distinct, meaning we get two different real numbers as solutions).
• If the discriminant is zero, there is exactly one real solution (a repeated root: the same real number appears as a solution twice).
• If the discriminant is negative, there are exactly 2 complex solutions (they are distinct, and they are complex conjugates).

(You can learn more about what the solutions of a quadratic formula represent in my article here.)

Remember that two complex conjugates have the form a + bi and a – bi.  In other words, they have the same real part, but opposite imaginary parts.

For example, 2 + 3i and 2 – 3i are complex conjugates.

Conclusion

Now you know when a quadratic equation has exactly one solution.  You also know what to look out for in terms of the discriminant, the graph, and the coefficients.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

You can learn more about other methods for solving quadratics in this resource from Lamar University.

You can learn more about quadratics in my other articles about quadratics with no real solution and quadratics with real solutions.

You might also want to read my article on when to use the quadratic equation.

This article goes into detail on how to use a quadratic to find the nature of the solutions (real or complex) of a cubic function.

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~Jonathon