SOLUTION: True or False? Explain your answer briefly. (a) For any real number c , the quadratic equation x^2 + x - c^2 = 0 has two distinct (real) solutions. (b) If a

SOLUTION: True or False? Explain your answer briefly. (a) For any real number c , the quadratic equation x^2 + x - c^2 = 0 has two distinct (real) solutions. (b) If a > 4 , then the equati

Question 1122567: True or False? Explain your answer briefly.
(a) For any real number c , the quadratic equation x^2 + x - c^2 = 0 has two distinct (real) solutions.
(b) If a > 4 , then the equation ax^2 + 4x + 1 = 0 has no (real) solutions.
(c) If b^2 − 4ac ≥ 0, then the quadratic equation ax^2 + bx + c = 0 has at most one solution.
Help with this question would be greatly appreciated!
Found 4 solutions by greenestamps, josmiceli, ikleyn, Alan3354:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

For the quadratic equation

the discriminant is

It is called the DISCRIMINant because it DISCRIMINates between three cases:

(1) If the discriminant is positive, there are 2 real solutions
(2) If the discriminant is zero, there is exactly 1 real solution
(3) If the discriminant is negative, there are no real solutions

(a) The discriminant is . What kinds of values (positive, zero, or negative) can that expression have?

(b) The discriminant is . Given that a > 4, what kinds of values can that expression have?

(c) The answer follows directly from the definition of the discriminant.

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Reply to the student's question....

In the standard form of the quadratic equation, , the "c" is whatever the constant term is.

In part (a) of your question, the constant term is "-c^2". That "c" is different than the "c" in the standard form of the quadratic equation. It just happens in this case that the constant term contains the variable "c".

Answer by josmiceli(19441)   (Show Source): You can put this solution on YOUR website!
(a)

2 real solutions
��-
(b)
The discrimination is

If , then

And

So, the discrimination is negative
and the solutions are imaginary ( not real )
��
(c)
If , then there is 1 real solution
which is
But if there are 2 real solutions

Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.
Be aware :   The solution by @josmiceli related to the part a) - is INCORRECT and IRRELEVANT.

Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
All quadratics have 2 solutions.
If the discriminant = 0, then the 2 solutions are equal.
Likewise, all cubics have 3 solutions, quartics have 4, etc.
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