Question

6x³-24x=0 25x³-10x²+x=0 x³-4x²-9x+36=0

Answer 1

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Answer 2

To solve each of these cubic equations, we can use various methods like factoring, the rational root theorem, or numerical methods. Let's solve each of the equations one by one:

1. 6x³ - 24x = 0

Step 1: Factor out the common term x from the equation: x(6x² - 24) = 0

Step 2: Set each factor to zero and solve for x: x = 0 (from the first factor) 6x² - 24 = 0

Step 3: Solve the quadratic equation: 6x² - 24 = 0 Divide the equation by 6 to simplify it: x² - 4 = 0 Now, add 4 to both sides: x² = 4 Take the square root of both sides: x = ±√4 x = ±2

So the solutions to the equation are x = 0, x = 2, and x = -2.

2. 25x³ - 10x² + x = 0

Step 1: Factor out the common term x from the equation: x(25x² - 10x + 1) = 0

Step 2: Set each factor to zero and solve for x: x = 0 (from the first factor) 25x² - 10x + 1 = 0

Step 3: Solve the quadratic equation. However, the quadratic term doesn't factor easily, so let's use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a where a = 25, b = -10, and c = 1

x = (10 ± √((-10)² - 4 * 25 * 1)) / 2 * 25 x = (10 ± √(100 - 100)) / 50 x = (10 ± √0) / 50 x = (10 ± 0) / 50

There are two solutions: x = 10/50 = 1/5 and x = 10/50 = 1/5.

So the solutions to the equation are x = 0, x = 1/5, and x = 1/5.

3. x³ - 4x² - 9x + 36 = 0

Step 1: Check if x = 1 is a root (meaning if it satisfies the equation): 1³ - 4 * 1² - 9 * 1 + 36 = 1 - 4 - 9 + 36 = 24 ≠ 0 Since x = 1 is not a root, it means that (x - 1) is not a factor.

Step 2: We can try to find rational roots using the rational root theorem. The possible rational roots are factors of the constant term (36) divided by factors of the leading coefficient (1). So the possible rational roots are ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, and ±36.

By trying these possible rational roots, we find that x = 4 is a root. Therefore, (x - 4) is a factor.

Step 3: Use polynomial long division or synthetic division to factor out (x - 4) from the equation: (x³ - 4x² - 9x + 36) ÷ (x - 4) = x² - 9

Step 4: Now we have a quadratic equation, which we can solve using the quadratic formula or factoring: x² - 9 = 0 (x + 3)(x - 3) = 0

Step 5: Set each factor to zero and solve for x: x + 3 = 0 --> x = -3 x - 3 = 0 --> x = 3

So the solutions to the equation are x = -3, x = 3, and x = 4.