The given equation is a cubic equation. According to Vieta's formulas, the sum of the roots of a cubic equation ax³ + bx² + cx + d = 0 is -b/a. In this case, a = 1, b = -9, so the sum of the roots a + b + c = -(-9) = 9.

The symbol [x] denotes the greatest integer less than or equal to x. Since a, b, and c are distinct real roots and their sum is 9, the maximum possible value of [a] + [b] + [c] is when a, b, and c are all integers and their sum is 9. In this case, [a] + [b] + [c] = a + b + c = 9.

However, the problem states that λ ∈ (-20, -18). The constant term of the cubic equation, λ, is equal to -abc according to Vieta's formulas. Since λ is negative, abc must also be negative, which means that one of a, b, and c is negative.

Therefore, the maximum possible value of [a] + [b] + [c] is less than 9. Since [x] is an integer for all x, the maximum possible value of [a] + [b] + [c] is 8.

Question

The given equation is a cubic equation. According to Vieta's formulas, the sum of the roots of a cubic equation ax³ + bx² + cx + d = 0 is -b/a. In this case, a = 1, b = -9, so the sum of the roots a + b + c = -(-9) = 9.

The symbol [x] denotes the greatest integer less than or equal to x. Since a, b, and c are distinct real roots and their sum is 9, the maximum possible value of [a] + [b] + [c] is when a, b, and c are all integers and their sum is 9. In this case, [a] + [b] + [c] = a + b + c = 9.

However, the problem states that λ ∈ (-20, -18). The constant term of the cubic equation, λ, is equal to -abc according to Vieta's formulas. Since λ is negative, abc must also be negative, which means that one of a, b, and c is negative.

Therefore, the maximum possible value of [a] + [b] + [c] is less than 9. Since [x] is an integer for all x, the maximum possible value of [a] + [b] + [c] is 8.

Knowee AI · Accepted Answer

The given equation is a cubic equation. According to Vieta's formulas, the sum of the roots of a cubic equation ax³ + bx² + cx + d = 0 is -b/a. In this case, a = 1, b = -9, so the sum of the roots a + b + c = -(-9) = 9.

The symbol [x] denotes the greatest integer less than or equal to x. Since a, b, and c are distinct real roots and their sum is 9, the maximum possible value of [a] + [b] + [c] is when a, b, and c are all integers and their sum is 9. In this case, [a] + [b] + [c] = a + b + c = 9.

However, the problem states that λ ∈ (-20, -18). The constant term of the cubic equation, λ, is equal to -abc according to Vieta's formulas. Since λ is negative, abc must also be negative, which means that one of a, b, and c is negative.

Therefore, the maximum possible value of [a] + [b] + [c] is less than 9. Since [x] is an integer for all x, the maximum possible value of [a] + [b] + [c] is 8.

Let equation x3 – 9x2 + 24x + λ = 0 has three real and distinct roots a, b, c. If λ ∈ (–20, –18), then the value of ([a] + [b] + [c]) will be

Question

Let equation $x^3 - 9x^2 + 24x + \lambda = 0$ has three real and distinct roots $a, b, c$ . If $\lambda \in (-20, -18)$ , then the value of $[a] + [b] + [c]$ will be

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