the roots of the equation
(ab-1)x²+(a+b)x-ab=0 are always
a) real
b) equal
c) complex
d) cannot be determined
You have to look at the discriminant b^2 - 4ac of the general equation formula
(a+b)^2 - 4(ab - 1)(-ab)
= a^2 + 2ab + b^2 + 4a^2 b^2 - 4ab
= a^2 - 2ab + b^2 + 4a^2b^2
= (a-b)^2 + 4a^2b^2
both terms clearly are both positive, so the discriminant is positive
Therefore you will have real roots
To determine the nature of the roots of the equation (ab-1)x²+(a+b)x-ab=0, we can use the discriminant. The discriminant is calculated as b² - 4ac for a quadratic equation in the form ax² + bx + c = 0.
Considering our equation, the coefficients are:
a = (ab-1)
b = (a+b)
c = -ab
Let's calculate the discriminant:
b² - 4ac = (a+b)² - 4(ab-1)(-ab)
Expanding the terms:
= a² + 2ab + b² + 4a²b² - 4a² - 4ab + 4ab - 4
Simplifying:
= a² + 2ab + b² + 4a²b² - 4a² - 4
Further simplification:
= 5a²b² + 2ab - 4a² - 4
Since we don't have any specific values for a and b, we cannot definitively determine whether the discriminant is positive, negative, or zero. This means we cannot determine the nature of the roots of the equation.
Therefore, the answer is option d) cannot be determined.
To determine the nature of the roots of the given equation, we can use the discriminant of the quadratic equation. The discriminant is given by the formula: Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
In this case, the given equation is (ab-1)x² + (a+b)x - ab = 0, which implies:
a = (ab-1)
b = (a+b)
c = -ab
We can substitute these values into the formula to find the discriminant:
Δ = [(a+b)²] - 4(ab-1)(-ab)
Expanding the equation and simplifying, we get:
Δ = (a² + 2ab + b²) + 4a²b² + 4ab
Now, to determine the nature of the roots based on the discriminant:
1. If Δ > 0, then the equation has two distinct real roots.
2. If Δ = 0, then the equation has two equal real roots.
3. If Δ < 0, then the equation has two complex roots.
4. If the coefficients a, b, and c are such that the discriminant cannot be determined, then the answer is "d) cannot be determined".
Since we have the expression for the discriminant, we can evaluate it based on the given values of a and b to determine the answer to the question.