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Which of the following has two distinct roots?
- a)x2+x−5=0
- b)x2+x+5=0
- c)none of these
- d)5x2−3x+1=0
Correct answer is option 'A'. Can you explain this answer?
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Which of the following has two distinct roots?a)x2+x−5=0b)x2+x+5...
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Which of the following has two distinct roots?a)x2+x−5=0b)x2+x+5...
To determine which of the given options has two distinct roots, we need to determine the discriminant of each equation. The discriminant is the part of the quadratic formula inside the square root, and it helps us determine the nature of the roots of a quadratic equation.
The quadratic equation is of the form ax^2 + bx + c = 0, where a, b, and c are constants.
The discriminant, denoted by Δ, is given by the formula Δ = b^2 - 4ac.
If Δ > 0, the equation has two distinct real roots.
If Δ = 0, the equation has two equal real roots.
If Δ < 0,="" the="" equation="" has="" no="" real="" />
Let's calculate the discriminant for each option:
a) x^2 + x + 5 = 0
In this case, a = 1, b = 1, and c = 5.
Δ = (1)^2 - 4(1)(5) = 1 - 20 = -19 (negative)
Since the discriminant is negative, this equation has no real roots.
b) x^2 + x * 5 = 0
In this case, a = 1, b = 5, and c = 0.
Δ = (5)^2 - 4(1)(0) = 25 - 0 = 25 (positive)
Since the discriminant is positive, this equation has two distinct real roots.
c) 5x^2 + 3x + 1 = 0
In this case, a = 5, b = 3, and c = 1.
Δ = (3)^2 - 4(5)(1) = 9 - 20 = -11 (negative)
Since the discriminant is negative, this equation has no real roots.
Therefore, the only option that has two distinct roots is option 'A' (x^2 + x + 5 = 0).
The quadratic equation is of the form ax^2 + bx + c = 0, where a, b, and c are constants.
The discriminant, denoted by Δ, is given by the formula Δ = b^2 - 4ac.
If Δ > 0, the equation has two distinct real roots.
If Δ = 0, the equation has two equal real roots.
If Δ < 0,="" the="" equation="" has="" no="" real="" />
Let's calculate the discriminant for each option:
a) x^2 + x + 5 = 0
In this case, a = 1, b = 1, and c = 5.
Δ = (1)^2 - 4(1)(5) = 1 - 20 = -19 (negative)
Since the discriminant is negative, this equation has no real roots.
b) x^2 + x * 5 = 0
In this case, a = 1, b = 5, and c = 0.
Δ = (5)^2 - 4(1)(0) = 25 - 0 = 25 (positive)
Since the discriminant is positive, this equation has two distinct real roots.
c) 5x^2 + 3x + 1 = 0
In this case, a = 5, b = 3, and c = 1.
Δ = (3)^2 - 4(5)(1) = 9 - 20 = -11 (negative)
Since the discriminant is negative, this equation has no real roots.
Therefore, the only option that has two distinct roots is option 'A' (x^2 + x + 5 = 0).
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