4x^2 – 5x – 12 = 0 Introduction Amidst the vast realm of mathematical intricacies and polynomial mysteries emerges the…

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To solve the equation 4x^2 – 5x – 12 = 0, we can use either the factoring method or the quadratic formula.

**Method 1: Factoring** Step 1: Write the equation in the form ax^2 + bx + c = 0 4x^2 – 5x – 12 = 0

Step 2: Factor the quadratic expression (4x + 3)(x – 4) = 0

Step 3: Set each factor equal to zero and solve for ‘x’ 4x + 3 = 0 or x – 4 = 0 4x = -3 or x = 4 x = -3/4 or x = 4

**Method 2: Quadratic Formula** The quadratic formula is given by: x = (-b ± √(b^2 – 4ac)) / 2a

For our equation, where a = 4, b = -5, and c = -12: x = (5 ± √(5^2 – 4 * 4 * -12)) / 2 * 4 x = (5 ± √(25 + 192)) / 8 x = (5 ± √217) / 8

So the two solutions for ‘x’ are: x = (5 + √217) / 8 x = (5 – √217) / 8

Therefore, the solutions to the equation 4x^2 – 5x – 12 = 0 are approximately: x ≈ 2.553 and x ≈ -0.553