Quadratic Formula: Positive or Negative Root?

In summary, the quadratic formula yields two roots, one when adding the positive square root and one when subtracting it. It does not matter if the number under the square root is a perfect square or not. Also, it is incorrect to use the plus/minus sign when finding the square root, as \sqrt{x} denotes the positive square root.
  • #1
Holocene
237
0
Regarding [tex]\displaystyle{\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}[/tex]

How should "[tex]\displaystyle{\pm}[/tex]" be treated?

I know a square root can be both possitive and negative, but does the quardratic forumla dictate that a possitive root should be added to -b, or does it dictate that a postive root should be subtracted from -b?
 
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  • #2
The quadradic forumal yields 2 roots. You do both.
 
  • #3
Some people get confused because other times they see the plus/minus sign, they have to choose correctly, whilst in this case you do it both. Thats why you may also see the formula written as;

[tex]x_1= \displaystyle{\frac{-b + \sqrt{b^2 - 4ac}}{2a}}[/tex]

[tex]x_2 = \displaystyle{\frac{-b - \sqrt{b^2 - 4ac}}{2a}}[/tex]
 
  • #4
Integral said:
The quadradic forumal yields 2 roots. You do both.

Sorry, I'm a little confused.

Say a particular solution to a quadratic equation is [tex]\displaystyle{\frac{2 \pm \sqrt{7}}{3}}[\tex]

If 7 was a perfect sqaure, would the root get added to or subtrected from 2?
 
  • #5
Holocene said:
Sorry, I'm a little confused.

Say a particular solution to a quadratic equation is [tex]\frac{2 \pm \sqrt{7}}{3}[/tex]

If 7 was a perfect square, would the root get added to or subtracted from 2?

It doesn't matter if 7 is a perfect square or not (and it isn't). There are TWO roots to a quadratic equation, hence the quadratic formula has TWO solutions. One of the solutions is when you add the square root part, the other solution is when you subtract it. Look at my previous post.
 
  • #6
Gib Z said:
It doesn't matter if 7 is a perfect square or not (and it isn't). There are TWO roots to a quadratic equation, hence the quadratic formula has TWO solutions. One of the solutions is when you add the square root part, the other solution is when you subtract it. Look at my previous post.

Okay I've got it. The 2 solutions are adding a possitive root, and subtracting a possitive root, correct? (Subtracting a negative root is = to adding it, adding a negative root is = to subtracting it).
 
  • #7
Holocene said:
Okay I've got it. The 2 solutions are adding a possitive root, and subtracting a possitive root, correct? (Subtracting a negative root is = to adding it, adding a negative root is = to subtracting it).
[itex]\sqrt{x}[/itex] is used to denote the positive square root of x. Thus [itex]\sqrt{4}=2[/itex] is correct while [itex]\sqrt{4}=\pm2[/itex] is incorrect. Thinking that the latter is correct is too common a mistake - make sure you aren't making it.
 
  • #8
morphism said:
[itex]\sqrt{x}[/itex] is used to denote the positive square root of x. Thus [itex]\sqrt{4}=2[/itex] is correct while [itex]\sqrt{4}=\pm2[/itex] is incorrect. Thinking that the latter is correct is too common a mistake - make sure you aren't making it.

Got it, thanks.
 

1. What is the quadratic formula?

The quadratic formula is a mathematical formula used to solve quadratic equations, which are equations in the form ax^2 + bx + c = 0. It gives the solutions for the values of x that make the equation true.

2. How do you determine whether the quadratic formula will have a positive or negative root?

The discriminant, which is the part of the quadratic formula under the square root symbol (b^2 - 4ac), determines whether the quadratic equation has two real solutions (positive and negative roots), one real solution (when the discriminant is 0), or no real solutions (when the discriminant is negative).

3. What is a positive root in the context of the quadratic formula?

A positive root is one of the solutions to the quadratic equation that gives a positive value for x. This means that when the positive root is substituted into the equation, it makes the equation true.

4. Is it possible for the quadratic formula to have only a negative root?

Yes, it is possible for the quadratic formula to have only a negative root. This occurs when the discriminant is negative, indicating that the equation has no real solutions. In this case, the negative root is considered to be an imaginary number.

5. Can the quadratic formula be used to solve any type of equation?

No, the quadratic formula can only be used to solve quadratic equations, which are equations in the form ax^2 + bx + c = 0. It cannot be used for linear equations (in the form ax + b = 0) or cubic equations (in the form ax^3 + bx^2 + cx + d = 0).

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