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If the discriminant is greater than zero (b^2 - 4ac) > 0, then the equation have two roots that are real and unequal. Further, the roots are rational if and only if (b^2 - 4ac) is a perfect square, otherwise the roots are irrational.
Example:
Find the equation whose roots are x = u/v and x = v/u
Solution:
x = u/v
x - u/v = 0
x = v/u
x - v/u = 0
Therefore:
(x - u/v)(x - v/u) = (0)(0) or
(x - u/v)(x - v/u) = 0
Let c = u/v and d = v/u. We can write this equation in equation in the form of:
(x - c)(x - d) = 0
x^2 - cx - dx + CD = 0 or
x^2 - (c +d)x + CD = 0
The sum of the roots is:
c + d = u/v + v/u = (u)(u)/(v)(u) + (v)(v)/(u)(v) = u^2/uv + v^2/uv = (u^2 + v^2)/uv
The product of the roots is:
(c)(d) = (u/v)(v/u) = uv/vu = uv/uv = 1
Substitute the sum and the product of the roots into the formula, and we'll have:
x^2 - (c +d)x + CD = 0
x^2 - [(u^2 + v^2)/uv]x + 1 = 0 Multiply both sides of the equation by uv
(uv)[x^2 - ((u^2 + v^2)/uv))x + 1] = (uv)(0)
(uv)x^2 - (u^2 + v^2)x + uv = 0 which is the equatiopn whose roots are u/v, v/u
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How many solutions are there if the discriminant of the equation is positive?
There are two distinct real solutions.
What type of equation is b2-4ac?
6
How many real solutions does a quadratic equation have if its discriminant is zero?
It will then have two equal real solutions
A positive discriminant means the quadratic equation will have how many solutions?
Two distinct real solutions.
How many real solutions does the following quadratic equation have by using the discriminant?
If the discriminant > 0 then 2 distinct real solutions.If the discriminant = 0 then 1 double real solution.If the discriminant < 0 then no real solutions (though there are two complex solutions).
What does the discriminant tell you?
The discriminant tells you how many solutions there are to an equation The discriminant is b2-4ac For example, two solutions for a equation would mean the discriminant is positive. If it had 1 solution would mean the discriminant is zero If it had no solutions would mean that the discriminant is negative
How many solutions are there if the discriminant of the equation is positive?
There are two distinct real solutions.
How do you know if a quadratic equation will have more than one solutions?
Write the quadratic equation in the standard form: ax2 + bx + c = 0 Then calculate the discriminant = b2 - 4ac If the discriminant is greater than zero, there are two distinct real solutions. If the discriminant is zero, there is one real solution. If the discriminany is less than zero, there are no real solutions (there will be two distinct imaginary solutions).
Why are there usually two solutions in quadratic equations and when do they only have one solution?
If the discriminant of the quadratic equation is greater than zero then it will have two different solutions. If the discriminant is equal to zero then it will have two equal solutions. If the discriminant is less than zero then it will have no real solutions.
If the discriminant is negative the equation has solutions?
As stated in the attached link, there are three possible discriminant conditions: Positive, Zero, or Negative. If the discriminant is negative, there are no real solutions but there are two imaginary solutions. So, yes there are solutions if the discriminant is negative. The solutions are imaginary, which is perfectly acceptable as solutions.
If the discriminant of an equation is negative?
It has two complex solutions.
What type of equation is b2-4ac?
6
How many real solutions does a quadratic equation have if its discriminant is zero?
It will then have two equal real solutions
A positive discriminant means the quadratic equation will have how many solutions?
Two distinct real solutions.
How many real solutions does the following quadratic equation have by using the discriminant?
If the discriminant > 0 then 2 distinct real solutions.If the discriminant = 0 then 1 double real solution.If the discriminant < 0 then no real solutions (though there are two complex solutions).
If the discriminant of an equation is positive.What is true of the equation?
The equation has two real solutions.
How can you have one solution in something that is quadratic?
b^2 - 4ac, the discriminant will tell you that a quadratic equation may have one real solution( discriminant = 0 ) , two real solutions( discriminant > 0 ), or no real solutions( discriminant < 0 ).
