SOLUTION: This is one main question with several parts. Please help. Thank you, How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equa

Question 173956: This is one main question with several parts. Please help. Thank you,
How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain.
Found 3 solutions by vleith, solver91311, Alan3354:
Answer by vleith(2983)   (Show Source): You can put this solution on YOUR website!
1) Look at the value of the discrimenent. Then you can see if there are 0, 1 or 2
2) If you are given roots, then you can 'recreate' the eqaution as follows
y = (x - X1)(x - X2) sub in the roots for X1 and X2, then expand the equation
3) Sure, imagine a parabola that intersects the x axis in two places. Now imagine a different parabola, with the same two intercepts but a more or less steep curve and a vertex higher of lower.
Easier to draw it than describe it.
Layout a coordinate plane
Pick two points on the x axis
pick a vertex below the x axis
Draw the parabola that connects all three of those points.
Now, move the vertex point twice as far below the x axis as the first vertex.
Now, draw the parabola that contains the new vertex and the 2 roots on the x axis.
Now ou see two different curves with the same roots.
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!
I don't think you stated your initial question precisely. I think what you meant to ask is "How do you know if a quadratic equation will have one, two, or no real number solutions?

In fact, the fundamental theorem of algebra tells us that a polynomial of degree n has n roots. Therefore a quadratic, which is a polynomial of degree 2, must ALWAYS have two roots. However, sometimes the roots are complex numbers so when you restrict your definition of 'solution' to the real numbers, you can have a 'no solution' result.

As to the question about one root, that is still a subject of some debate. Some say that a quadratic always has two roots, but in the case of a perfect square trinomial, those two roots will be identical. Others say that there is a single root with a multiplicity of two for that situation.

The process to determine the character of the roots of a quadratic is as follows:
Step 1: Put the quadratic into standard form, namely

Step 2: Calculate the discriminant , which is the expression under the radical in the quadratic formula , namely: .

If then you either have two real and identical roots, or you have one root with a multiplicity of two, depending on your school of thought. This is your 'one solution' situation.

If , then you have two different real number roots. This is your 'two solutions' situation.

If , then you have a conjugate pair of complex number roots of the form where is the imaginary number defined by . This is your 'no solution' situation.

***********************

Given two numbers and , you can derive a quadratic polynomial with those two numbers as roots by using the fact that is a root of a polynomial in if and only if is a factor of the polynomial.

Therefore, given and as the two roots of a quadratic, one of the quadratic polynomials with and as roots can be derived from .

***********************

Your last question, "Is it possible to have different quadratic equations with the same solution?"

You need to be very careful with your language here.

has a solution set {-2,3}, but so does . However, one could successfully argue that these two equations are equivalent, and therefore are not really different quadratic equations. So, the answer to your question as stated is no, you can't have different quadradic EQUATIONS with the same solution.

But do not confuse a quadratic equation with a polynomial function of degree 2: . Looking at the previous example, and are two very different functions -- one is a parabola opening upwards and the other is a parabola opening downwards.
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain.
------------------
A quadratic will always have 2 solutions.
However, sometimes the 2 solutions are the same. For example:
x^2 -2x + 1 = 0
(x-1)*(x-1) = 0
x = 1
x = 1
It's the same point, (1,0), so some will say it's one solution. You can get into semantics.
Also, sometimes the solutions will be complex numbers, instead of real.
For example:
x^2 + 4 = 0

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

The discriminant -16 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

In the field of imaginary numbers, the square root of -16 is + or - .

The solution is , or
Here's your graph:

This has no REAL number solutions, but still has 2 solutions. You can see it doesn't cross the x-axis.
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There are an infinite number of quadratics with the same 2 solutions. 2 points don't define a parabola (to put in graphic terms), 3 are required to define the curve AFTER it's defined to be a parabola.

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