# How can x 2 be negative

## PQ Formula for quadratic equations

**This section deals with the use of the PQ formula to solve quadratic equations. First of all, we will show you what a quadratic equation is and what the PQ formula is needed for. In addition to text explanations, there are - as always - some examples for viewing.**

First of all, of course, the question arises: what is a quadratic equation? Well, it's an equation of the form ax^{2} + bx + c = 0 or an equation that can be reduced to this form. A, b and c are any numbers where a must be non-zero. Examples: 3x^{2} + 5x + 3 = 0 or x^{2} + 2x + 1 = 0.

In contrast to the equations that we have got to know so far (example: x + 5 = 0) there is still a quadratic component here. So how do you solve this equation for x? The answer to this question is PQ formula, which we want to explore in this section. Before that, however, I would like to point out the prior knowledge required. For those who still have problems with normal equations or the calculation of the roots, I recommend the following two articles. Everyone else can get started with the PQ formula right away.

Show:**Use the PQ formula**

To find an equation such as x^{2} + 2x + 1 = 0 to solve for x, we use the PQ formula in the following. First of all, I'll give you the formula and some general information. **Don't panic: a few examples explain this below**.

**How to solve a quadratic equation: **

- Put the equation in the form x
^{2}+ px + q = 0 - Find out "p" and "q"
- Plug this into the PQ formula
- Calculate the solution with it

So much for the theory. Time to clarify this with a few examples. Follow these examples using the 4-point list from above.

**Important NOTE**: In order not to confuse students with many fractions, some examples have been rounded.

**Example 1: **

**Explanations:** The "3" before the x^{2} disturbs! There must always be a "1", i.e. 1x^{2}. To achieve this, it is divided by 3. Then read p and q. The numbers of p and q are plugged into the PQ equation. Then the expression before and below the root is calculated. Then the root is taken from the value and it is added once and subtracted once.** A quadratic equation has a maximum of two real solutions. **

**Example 2:**

**Explanations:** The original task is already in the right form. Therefore, p and q can be determined equally. Then insert this into the equation and calculate it. As you can see from the result, there is twice the solution -2, i.e. x_{1} = -2 and x_{2} = -2.

**Note:**

Do you have an exam that also includes the PQ formula? Would you like to see if you can use this? Then you should still do our tasks / exercises on this topic.

**PQ formula: Negative root / observance of the sign**

There are two more small hints when calculating quadratic equations with the PQ formula from us:

- If you calculate the numbers under the root and then there is a negative number under the root, you can cancel. Then the equation has no solution (at least not for schoolchildren, students then have to do imaginary arithmetic).
- Pay attention to the sign! For example, do you have the task x
^{2}To solve -5x + 3 = 0, then p = -5. You then have to use this -5 in the PQ formula!

You can find an example for both cases here:

Only by carefully practicing tasks can you become confident in using the PQ formula. So we advise you to do our math solving quadratic equations exercises.

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