Number of solutions

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:

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.

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

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!

Not so bad once you see it in context!

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

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:

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?