The quadratic equation, a polynomial of degree 2, is represented by the equation 4×2 – 5x – 12 = 0. We have several ways we can use to answer this equation. Factoring is a method to divide the problem into components and locate the roots. The quadratic formula, which offers a formulaic response to any quadratic equation, is a different approach. Using these methods, we can identify the values of x that satisfy the equation. We can solve real-world problems involving quadratic relationships or locate critical points on a graph by resolving quadratic equations.
Understanding Quadratic Equations: An Introduction to 4x^2 – 5x – 12 = 0
Quadratic equations are fundamental in mathematics and have many uses in various disciplines. One example is 4×2 – 5x – 12 = 0, where x represents an unidentified variable.
The general form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are constants. The coefficients in this instance are 4 for a, 5 for b, and 12 for c.
The maximum x power is 2, according to the second-degree polynomial equation 4×2 – 5x – 12 = 0. We must identify the values of x that satisfy this equation to locate its solutions or roots.
Several ways to solve quadratic equations include factoring, completing the square, and utilizing the quadratic formula. These methods offer methodical approaches.
By investigating and comprehending quadratic equations such as 4×2 – 5x – 12 = 0, we can learn important lessons about the fundamentals of algebra and acquire problem-solving abilities that are useful across a wide range of mathematics and beyond.
Factoring the Quadratic Equation: Unraveling the Roots of 4x^2 – 5x – 12 = 0
Factoring is an effective technique for finding solutions to quadratic equations by dividing a quadratic equation into smaller parts. Applying this strategy, let’s find the roots of the equation 4×2 – 5x – 12 = 0.
We look for two binomial expressions that, when multiplied, give the original quadratic equation to factor it. Find two values that add up to -5 (the coefficient of x) and multiply to provide -48 (the product of the coefficients of x2 and the constant term).
We find that -8 and 6 satisfy the requirements by examining the factors of -48 and testing several combinations. As a result, the quadratic equation can now be written as (4x + 6)(x – 8) = 0.
We discover two possible answers by setting each number to zero: 4x + 6 = 0 and x – 8 = 0. We resolve these equations separately at x = -3/2 and x = 8.
As a result, the quadratic equation 4×2 – 5x – 12 = 0 can be factored into (4x + 6)(x – 8) = 0, which reveals the roots x = -3/2 and x = 8. By factoring, we can discover these answers and learn more about how quadratic equations behave.
Quadratic Formula: Solving 4x^2 – 5x – 12 = 0 with Mathematical Precision
The quadratic formula is an effective technique that enables us to answer any quadratic equation precisely. Let’s use this equation to determine the roots of 4×2 – 5x – 12 = 0.
According to the quadratic formula, the solutions for x can be determined using the formula: x = (-b (b2 – 4ac)) / (2a) for an equation of the type ax2 + bx + c = 0.
A = 4, B = 5, and C = 12 are the values for our equation. We can get the answers for x by entering these values as substitutes in the quadratic formula.
The calculation gives us the following result: x = (-(-5) ((-5)2 – 4 * 4 * -12)) / (2 * 4)
x = (5 ± √(25 + 192)) / 8 x = (5 ± √217) / 8
Therefore, (5 + 217) / 8 and (5 – 217) / 8 are the answers to the quadratic equation 4×2 – 5x – 12 = 0.
We can precisely solve quadratic equations using the quadratic formula, leading to accurate solutions. This formula offers a systematic way to identify the roots of any quadratic equation, allowing us to examine and comprehend their characteristics.
Analyzing the Solutions: Exploring the Nature and Significance of the Roots of 4x^2 – 5x – 12 = 0
Using factoring and the quadratic formula, we solved the quadratic equation 4×2 – 5x – 12 = 0. Let’s now examine these roots’ characteristics and importance. 4x ^ 2 – 5x – 12 = 0
The obtained roots are (5 + 217)/8 and (5 – 217)/8. We can comprehend the behaviour of the equation by looking at the roots’ characteristics.
The roots of the equation are real numbers, which is the first clue that they are real solutions. This suggests that the equation’s graph and the x-axis coincide at these locations.
Second, the roots are neither integers nor rational numbers but irrational quantities involving the square root 217. This indicates that it may be challenging to represent the solutions in straightforward numerical terms.
The roots also have two distinct signals, one positive and one negative. This shows where the x-axis is crossed twice by the parabolic curve defined by the equation.
4×2 – 5x – 12 = 0 solutions are examined to understand better the nature of the equation and its graphical representation. These roots help us understand the behaviour of quadratic functions by supplying details on the points of intersection.
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