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## 4x^2 – 5x – 12 = 0

**Introduction** Amidst the vast realm of mathematical intricacies and polynomial mysteries emerges the enigmatic quadratic equation “4x^2 – 5x – 12 = 0”. Various methodologies lay at our disposal to fathom the depths of this equation. Employing factoring, we disentangle its components and unearth its roots. Alternatively, the quadratic formula opens a gateway to obtain a formulaic response for any quadratic equation. Through these avenues, we unveil the elusive values of ‘x’ that satisfy the equation, enabling us to tackle real-world challenges, grasp quadratic relationships, and discern critical points in graphical representations.

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

**Comprehending Quadratic Equations**

**A Prelude to 4x^2 – 5x – 12 = 0** Quadratic equations, a fundamental facet of mathematics, find multifarious applications across diverse disciplines. Let us take 4x^2 – 5x – 12 = 0 as an illustrative example, wherein ‘x’ denotes an enigmatic variable. Represented in the general form of ax^2 + bx + c = 0, we encounter the constants 4 (a), 5 (b), and 12 (c) in this particular instance. As a second-degree polynomial equation, 4x^2 – 5x – 12 = 0 holds the highest power of ‘x’, prompting us to explore the values of ‘x’ that fulfill its conditions and unveil its roots. Factoring, completing the square, and the quadratic formula constitute several methodologies for resolving quadratic equations, each offering a methodical approach. By unraveling and grasping the essence of quadratic equations, such as 4x^2 – 5x – 12 = 0, we glean profound insights into the bedrock of algebra and foster problem-solving skills applicable across an extensive spectrum of mathematical conundrums and beyond.

**Decoding the Quadratic Equation**

** Unveiling the Roots of 4x^2 – 5x – 12 = 0 through Factoring** Factoring proves an efficacious technique for ascertaining the solutions to quadratic equations by partitioning them into manageable components. Let us harness this strategy to extract the roots of equation 4x^2 – 5x – 12 = 0. Our quest involves unearthing two binomial expressions whose multiplication yields the original quadratic equation. To achieve this, we seek two values whose sum amounts to -5 (the coefficient of ‘x’) and whose product equals -48 (the product of the coefficients of x^2 and the constant term). After scrutinizing the factors of -48 and exploring various combinations, we discern that -8 and 6 satisfy the stipulations. Consequently, the quadratic equation transforms into (4x + 6)(x – 8) = 0. Setting each term to zero leads us to two possible answers: 4x + 6 = 0 and x – 8 = 0. Consequently, we deduce the solutions x = -3/2 and x = 8. By effectuating factoring, we unravel the enigmatic code 4x^2 – 5x – 12 = 0 into (4x + 6)(x – 8) = 0, thus unveiling the roots x = -3/2 and x = 8. This process endows us with insights into the behavior of quadratic equations.

**The Quadratic Formula**

** Precisely Solving 4x^2 – 5x – 12 = 0** The quadratic formula constitutes a potent tool, bestowing upon us the precise solutions to any quadratic equation. Let us wield this formula to unearth the roots of 4x^2 – 5x – 12 = 0. For an equation of the form ax^2 + bx + c = 0, the quadratic formula establishes that the solutions for ‘x’ are determined as follows: x = (-b ± √(b^2 – 4ac)) / (2a). In the case of our equation, we denote A = 4, B = 5, and C = 12. Employing these values as substitutions in the quadratic formula, we ascertain the answers for ‘x’. Thus, our calculation yields the following result: x = (5 ± √(25 + 192)) / 8, which simplifies to x = (5 ± √217) / 8. Accordingly, the quadratic equation 4x^2 – 5x – 12 = 0 culminates in the roots (5 + √217) / 8 and (5 – √217) / 8. Through the quadratic formula, we garner a methodical means to fathom the roots of any quadratic equation and discern their intrinsic attributes.

**Analyzing the Solutions**

** Delving into the Essence and Significance of the Roots of 4x^2 – 5x – 12 = 0** Having leveraged factoring and the quadratic formula to resolve the quadratic equation 4x^2 – 5x – 12 = 0, let us now scrutinize the nature and importance of these roots. The acquired roots amount to (5 + √217) / 8 and (5 – √217) / 8. An analysis of these roots’ attributes allows us to grasp the behavior of the equation. These roots manifest as real numbers, signifying their status as real solutions. This observation underscores the intersection of the equation’s graph with the x-axis at these particular locations. Furthermore, the roots constitute irrational quantities entailing the square root of 217, which implies that their representation in straightforward numerical terms might prove challenging. Distinct signals, one positive and one negative, mark the roots. This characteristic highlights the two points of intersection where the equation’s parabolic curve intersects the x-axis. By examining the solutions to 4x^2 – 5x – 12 = 0, we garner insights into the behavior of quadratic functions and glean essential details concerning points of intersection.