# 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.

$$b^2-4ac>0$$ | Two real solutions |

$$b^2-4ac=0$$ | One real solution |

$$b^2-4ac<0$$ | No real solutions |

## 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:

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:

## Study summary

- Rewrite the quadratic equation in standard form (if it’s not already).
- Identify the coefficients.
- Calculate the discriminant of the quadratic equation.
- 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:**

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

*Solutions:*

- $$2$$ real solutions
- $$2$$ real solutions
- $$1$$ real solution
- 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:

- Solve it and count the number of solutions.
- Compare the value of the discriminant to $$0$$.
- 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?