SOLUTION: <pre> 25 + 4x² = -20x discriminant how do you find the <s>determinant</s> then the number of rational, irrational or complex roots??</pre>

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: <pre> 25 + 4x² = -20x discriminant how do you find the <s>determinant</s> then the number of rational, irrational or complex roots??</pre>       Log On


   

Question 629356: 
25 + 4x² = -20x
                    discriminant
how do you find the determinant then the number of rational, irrational or complex roots??
Found 2 solutions by solver91311, Edwin McCravy:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


First put your equation into standard form,

then

Find the Discriminant, and evaluate the nature of the roots as follows:

No calculation quick look: If the signs on and are opposite, then guaranteed.

Two real and unequal roots. If is a perfect square, the quadratic factors over (the rationals).

One real root with a multiplicity of two. That is to say that the trinomial is a perfect square and has two identical factors. Presuming rational coefficients, the root will be rational as well.

A conjugate pair of complex roots of the form where is the imaginary number defined by

John

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Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
25 + 4x² = -20x

how do you find the determinant then the number of rational, irrational or complex roots??

The word is "discriminant", not "determinant".

First you get the quadratic equation in descending order with 0
on the right.

      25 + 4x² = -20x

4x² + 20x + 25 = 0

Compare that to the general quadratic equation:

 ax² + bx + c = 0

If you remember the quadratic formula, x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ 

then you already know the discriminant formula because it is
what is under the aquare root part of the quadratic formula.

Discriminant = b² - 4ac      

Comparing your equation to the general quadratic equation,

a = 4, b = 20, c = 25

Substitute in the discriminant formula:

Discriminant = b² - 4ac 

Discriminant = (20)² - 4(4)(25)

Discriminant = 400 - 400

Discriminant = 0

The rules are

1. If the discriminant is positive there are 2 real roots
   A.  If the discriminant is a perfect square, both roots are rational
   B.  If the discriminant is not a perfect square, both roots are irrational.
2. If the discriminant is negative, there are 2 conjugate complex roots.
3. If the discriminant is 0, there is just one real rational root.

Your problem is case 3, so there is one real rational root. 

Edwin