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Which one of the following is one of the roots of the equation (b – c) x2 + (c – a) x + (a – b) = 0 ?
- a)(c – a) / (b – c)
- b)(a – b) / (b – c)
- c)(b – c) / (a – b)
- d)(c – a) / (a – b)
Correct answer is option 'B'. Can you explain this answer?
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Explanation:
The given equation is a quadratic equation in the form ax^2 + bx + c = 0. We need to find one of the roots of this equation.
To find the roots of a quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing this with the given equation, we can determine the values of a, b, and c:
a = (b - c)
b = (c - a)
c = (a - b)
Substituting these values into the quadratic formula, we get:
x = (-(c - a) ± √((c - a)^2 - 4(b - c)(a - b))) / (2(b - c))
Simplifying this expression further:
x = (-(c - a) ± √(c^2 - 2ac + a^2 - 4ab + 4ac - 4b^2 + 4bc)) / (2(b - c))
x = (-(c - a) ± √(a^2 + b^2 - c^2 - 2ab + 4ac - 4b^2 + 4bc)) / (2(b - c))
x = (-(c - a) ± √(a^2 + b^2 + 4ac - 4b^2 + 4bc - c^2 - 2ab)) / (2(b - c))
x = (-(c - a) ± √((a^2 - 2ab + b^2) + 4(ac + bc - b^2))) / (2(b - c))
x = (-(c - a) ± √((a - b)^2 + 4(ac + bc - b^2))) / (2(b - c))
Since we are looking for one of the roots, we can choose the positive sign in front of the square root:
x = (-(c - a) + √((a - b)^2 + 4(ac + bc - b^2))) / (2(b - c))
Comparing this with the given options, we can see that option B, (a - b) / (b - c), matches the expression for x.
Therefore, option B is one of the roots of the equation.
The given equation is a quadratic equation in the form ax^2 + bx + c = 0. We need to find one of the roots of this equation.
To find the roots of a quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing this with the given equation, we can determine the values of a, b, and c:
a = (b - c)
b = (c - a)
c = (a - b)
Substituting these values into the quadratic formula, we get:
x = (-(c - a) ± √((c - a)^2 - 4(b - c)(a - b))) / (2(b - c))
Simplifying this expression further:
x = (-(c - a) ± √(c^2 - 2ac + a^2 - 4ab + 4ac - 4b^2 + 4bc)) / (2(b - c))
x = (-(c - a) ± √(a^2 + b^2 - c^2 - 2ab + 4ac - 4b^2 + 4bc)) / (2(b - c))
x = (-(c - a) ± √(a^2 + b^2 + 4ac - 4b^2 + 4bc - c^2 - 2ab)) / (2(b - c))
x = (-(c - a) ± √((a^2 - 2ab + b^2) + 4(ac + bc - b^2))) / (2(b - c))
x = (-(c - a) ± √((a - b)^2 + 4(ac + bc - b^2))) / (2(b - c))
Since we are looking for one of the roots, we can choose the positive sign in front of the square root:
x = (-(c - a) + √((a - b)^2 + 4(ac + bc - b^2))) / (2(b - c))
Comparing this with the given options, we can see that option B, (a - b) / (b - c), matches the expression for x.
Therefore, option B is one of the roots of the equation.
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