How can one solve quadratic functions / quadratic equation? This is exactly what we'll look at in the next few sections. The following content is offered:

• First of all there is Explanationswhat is a quadratic function / equation at all. And then it's about solving them
• It will Examples pre-calculated to show the ABC formula / midnight formula and PQ formula.
• tasks and Exercises enable you to practice solving quadratic functions yourself.
• Videos for solving quadratic functions are also available.

The first thing here is a brief explanation of what quadratic functions or quadratic equations are. And then it comes to solving examples. Another tip: If you have problems with the content, you may be missing a few important prior knowledge: In this case, please deal with the topics of solving linear equations and drawing functions.

Explanation: Solve quadratic functions / equations

For example, a quadratic function or equation can look like this:

How can one solve quadratic functions or quadratic equations? Well, there are two very typical ways to do this. And these are called the PQ formula and the ABC formula. We'll look at both of them now.

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Quadratic function: PQ formula and ABC formula

We have now got to know what quadratic equations are. Let's look at two methods of solving these. On the one hand there is the PQ formula. On the other hand, there is the midnight formula, which is sometimes also called the ABC formula. Let's start with the PQ formula.

Quadratic equation / function solution formula

In order to be able to use the PQ formula, one must first ensure that before x2 a 1 stands and the equation is brought to the form with = 0. Then you can read off p and q and simply insert them. First the equation for the solution, then examples.

Quadratic equation / function example 1:

To understand the example 3x2 + 9x + 5 = - 1 using the PQ formula. First the invoice, then the explanation:

And this is how it was calculated:

• First we have to bring everything to the form = 0, to do this we first have to calculate +1.
• We also need before the x2 a 1, i.e. 1x2 and not 3x like here2. By dividing by 3, we can do this.
• We read p and q and insert them into the solution equation.
• We calculate everything before the root and below the root.
• We still have a plus (+) and a minus (-) in front of the root. Now we determine x1 with the plus and x2 with the minus.
• So there are two solutions to x1 = -1 and x2 = -2.

Quadratic equation / function example 2:

Let's make another example, which should be solved with the PQ formula. The quadratic function / equation x is given2 -5x - 6 = 0.

Before x2 there is a 1 and the equation has the form with = 0. So we can read off p and q directly, insert them and calculate the two results. Tip: More examples in our article PQ formula.

Quadratic equation function with ABC formula:

In addition to the PQ formula, there is also the ABC formula, which is sometimes also called the midnight formula. Here, too, the equation is reduced to the form with = 0, but x2 also stand another number.

Let's take our equation from above, which we had already solved with the PQ formula, and now solve it with the midnight formula. First back to the form with = 0. Then we read off a, b and c and insert them into the solution equation of the ABC formula (midnight formula). And as you can see: Even with the midnight formula, we get the two solutions for x1 = -1 and x2 = -2.

Conclusion: PQ formula and ABC formula deliver identical results.

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Use PQ formula

In this video we are going to take a look at the PQ formula. This is presented with a solution formula and numerous examples. This is how you learn how to solve quadratic equations / functions.

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