When dealing with quadratic equations, we regularly see either two genuine solutions or two complicated solutions. However, the is also feasible that a quadratic will have specifically one solution.
You are watching: A quadratic equation has exactly one real number solution. which is the value of its discriminant?
So, once does a quadratic have one solution? A quadratic equation has actually one solution once the discriminant is zero. Native an algebra standpoint, this means b2 = 4ac. Visually, this means the graph the the quadratic (a parabola) will have its vertex resting on the x-axis.
Of course, we can additionally look in ~ the coefficients in a quadratic equation come tell if it has actually one solution.
In this article, fine talk around how you have the right to tell the a quadratic has one solution. We’ll additionally look at some examples, in addition to how to create a quadratic equation provided that it only has actually one solution.
Let’s gain started.
When walk A Quadratic have actually One Solution?
There room a few different means to tell as soon as a quadratic equation has one solution:Look at the discriminant – if the is zero, there is only one solution to the quadratic.Look in ~ the graph – if the peak of the parabola rests on the x-axis, there is just one systems to the quadratic.Look at the coefficients – there is a one-of-a-kind pattern that will tell you as soon as there is only one equipment to the quadratic (more ~ above this later on in the article!)
We’ll begin with a an approach that uses the discriminant.
Look in ~ The Discriminant
The first way to tell if a quadratic has actually one solution is to look at the discriminant. If the discriminant is zero, climate the quadratic equation has actually only one actual solution.
As you have the right to see in the graph pictured above, the crest (valley bottom) of this parabola lies on the x axis. This method the quadratic equation x2 – 4x + 4 has actually one genuine solution (at x = 2).
Be careful: for a quadratic equation to have only one real solution (a dual root), its graph must touch the x axis exactly once.
If the graph touches the x axis twice, then it has actually two unique real solutions.
If the graph does no touch the x axis at all, climate it has actually two complicated solutions (and no real solutions).
Look at The Coefficients
You can also look in ~ the coefficients that a quadratic equation in standard type to tell if it has actually one real solution. Remember the the standard form the a quadratic equation has zero top top one side, and terms in descending stimulate on the other:ax2 + bx + c = 0
There are two feasible cases to address here: either a = 1, or a is not equal come 1.When The Coefficient that x2 Is same To 1 (a = 1)
In this case, we look for c = b2 / 4. If this is true, then the quadratic has one solution, i beg your pardon is x = -b / 2.
For example, the quadratic equation x2 + 8x + 16 has actually 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 very same as the value of c. The one systems is:Px = -b / 2x = -8/2x = -4
We can likewise find this systems by factoring the quadratic as:x2 + 8x + 16 = (x + 4)(x + 4)
which gives us a systems of x = -4 (a actual repeated root). We can also use the quadratic formula through a = 1, b = 8, and also c = 16 to get the same an outcome of x = -4.When The Coefficient the x2 Is no Equal to 1 (a is no equal to 1)
In this case, divide the whole quadratic equation by a. Then, you room in the an initial case, as soon as the coefficient that x2 is same to 1.
For example, take into consideration the equation:2x2 + 12x + 18 = 0
Since a = 2, we will divide both sides by 2.
This leaves united state with:x2 + 6x + 9 = 0
For this new quadratic equation, we have a = 1, b = 6, and also c = 9.
Then we have:b2 / 4= 62 / 4= 36 / 4= 9
which is the very same as the worth of c. The one solution is x = -b / 2 = -6 / 2 = -3.
We can likewise find this equipment by factoring the quadratic as:x2 + 6x + 9 = (x + 3)(x + 3)
which provides us a equipment of x = -3 (a genuine repeated root). We can additionally use the quadratic formula with a = 1, b = 6, and also c = 9 to acquire the same an outcome of x = -3.
Remember the you can always use a calculator to assist you verify the remedies to a quadratic equation. Girlfriend can also use a quadratic equation solver, such together this one from WolframAlpha.
For WolframAlpha’s calculator, mental that:The quadratic coefficient (x2 coefficient) way aThe direct coefficient (x coefficient) means bThe constant coefficient means c
Examples that Quadratic Equations v One Solution
Here room some instances of quadratic equations through one solution. Look at them to check out 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 point you might an alert is the the x coefficients (b values) space all even numbers:2, 4, 6, 8, and also 10
These come from copy the numbers 1, 2, 3, 4, and 5.
Another thing you might notice is the the consistent terms (c values) room all perfect squares:1, 4, 9, 16, and 25
These come indigenous squaring the number 1, 2, 3, 4, and also 5.
One last point you might notice is that the remedies are the negatives of the numbers 1, 2, 3, 4, and 5.
After we notification the pattern, we check out the beauty beauty of it: you deserve to pick any whole number n and also create a quadratic with a solitary solution (the systems is n).
Here’s exactly how to carry out it.First, choose your entirety number n. Remember that a entirety number has actually no fountain or decimals. Because that this example, I pick n = 7.Next, calculate 2n. This will certainly be the worth of b. Because that this example, b = 2n = 2*7 = 14.Then, calculation n2. This will be the value of c. Because that this example, c = n2 = 72 = 49.Finally, her quadratic v one solution has actually a = 1, b = 2n, and c = n2. So, the quadratic would look favor x2 + 2n + n2. The solution will be –n (the an adverse of the totality number n you chose at the beginning).
For this example, our quadratic through one equipment is x2 + 14x + 49. The systems is x = -7.
We deserve to also adjust the sign of every the b values to obtain an entire new set that 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 kind of given quadratic equation v one solution and multiply by any type of number (except zero) to gain a new quadratic equation v the exact same solution.
How do You create A Quadratic Equation through One Solution?
Now it’s time come think around working backwards. That way taking one number and writing a quadratic equation whose just solution is the number.
This is simple to do. Even better, there are infinitely plenty of such equations for a given solution.
Let’s to speak we want to write a quadratic equation with just one solution. Let’s likewise assume that we desire the equipment to it is in the number n.
To perform this, we would merely write the equation:(x – n)(x – n) = 0
After using FOIL on the parentheses, us would obtain the equationx2 – 2nx + n2 = 0
This equation has precisely one solution, i beg your pardon is x = n (it is a dual real root).
Example: writing A Quadratic Equation through One actual Solution
For example, stop say I want a quadratic equation through a systems of n = 3.
Then the equation would be(x – 3)(x – 3) = 0
After using FOIL, us would gain the equationx2 – 6x + 9 = 0
This equation has exactly one solution, i beg your pardon is the value x = 3. We can additionally generate as many an ext equations together we choose by simply multiplying both political parties by 2, 3, 4, 5, and so forth:2x2 – 12x + 18 = 0 (multiplied through 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 constantly Have two Solutions?
In state of actual solutions, there are constantly either 0, 1, or 2 real solutions to a quadratic equation, depending upon the sign of the discriminant.
However, over there is constantly at the very least 1 equipment if you count both real and facility numbers.If the discriminant is positive, there are precisely 2 real solutions (they space distinct, definition we acquire two various real numbers together solutions).If the discriminant is zero, over there is exactly one genuine solution (a repeated root: the same real number appears as a solution twice).If the discriminant is negative, over there are specifically 2 facility solutions (they are distinct, and they are complicated conjugates).
(You can learn much more about what the services of a quadratic formula stand for in my article here.)
Remember that two facility conjugates have actually the kind a + bi and also a – bi. In various other words, they have actually the same real part, however opposite imagine parts.
For example, 2 + 3i and 2 – 3i are complex conjugates.
Now you understand when a quadratic equation has exactly one solution. You likewise know what come look out for in regards to the discriminant, the graph, and also the coefficients.
I hope you found this post helpful. If so, you re welcome share it with someone who have the right to use the information.
You can learn much more about other techniques for fixing quadratics in this resource from Lamar University.
You have the right to learn more about quadratics in my various other articles around quadratics v no real solution and quadratics with real solutions.
You might likewise want to review my post on as soon as to use the quadratic equation.
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This post goes into information on exactly how to use a quadratic to find the nature of the services (real or complex) of a cubic function.
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