Education
4x^2 – 5x – 12 = 0: Understanding and Solving Quadratic Equations
In many different areas of mathematics and science, quadratic equations are a common need. The most common form of such equations is 4x 2 5x 12 0, where a, b, and c are constants and x is the variable whose value must be determined. In this post, we will learn how to solve the quadratic equation 4×2 – 5x – 12 = 0.
Understanding Quadratic Equations
Because the greatest power of the variable ‘x’ is the square root (x2), quadratic equations are so named. Two actual solutions, one real solution, or no real solutions at all are possible for these equations. The discriminant, the number within the square root in the quadratic formula, determines the number and kind of solutions.
Solving Quadratic Equations
Solving quadratic equations often requires one of two approaches: factoring or using the quadratic formula.
Factoring Method
In order to factor an equation, you must rewrite it so that it can be factored into two binomial expressions. When the problem can be factored simply, this procedure is quite simple.
Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation. It is expressed as:
x = (-b ± √(b^2 – 4ac)) / (2a)
Where ‘a,’ ‘b,’ and ‘c’ are the coefficients from the equation ax^2 + bx + c = 0.
The Equation 4x^2 – 5x – 12 = 0
Let’s focus on the specific quadratic equation 4x^2 – 5x – 12 = 0. Our goal is to find the values of ‘x’ that satisfy this equation.
Solving 4x^2 – 5x – 12 = 0 using Factoring
Factoring the equation 4×2 – 5x – 12 = 0 into two binomial expressions allows us to use the factoring method to solve the equation. Since it might be difficult to factor this specific equation, the quadratic formula will be used instead.
Solving 4x^2 – 5x – 12 = 0 using the Quadratic Formula
Let’s use the quadratic formula to find the solutions of 4x^2 – 5x – 12 = 0. We have ‘a’ = 4, ‘b’ = -5, and ‘c’ = -12. Plugging these values into the formula:
x = (-(-5) ± √((-5)^2 – 4 * 4 * (-12))) / (2 * 4)
Simplifying further:
x = (5 ± √(25 + 192)) / 8
x = (5 ± √217) / 8
So, the solutions for 4x^2 – 5x – 12 = 0 are:
x₁ = (5 + √217) / 8 and x₂ = (5 – √217) / 8
Real-World Applications of Quadratic Equations
Quadratic equations are useful in many situations outside of academia. They find widespread use in fields as diverse as physics, engineering, economics, and daily life. They may be used to estimate the effects of supply and demand on pricing, for instance, or to calculate the path of a projectile, among other things.
Conclusion
The quadratic equation is an important tool in mathematics and has many real-world applications. Learning how to solve them, whether via factoring or the quadratic formula, is a talent that may be put to good use. In this post, we utilized the quadratic formula to show how to solve the problem 4x 2 5x 12 0.
FAQs
Q1: What is a quadratic equation?
A1: A quadratic equation is a second-degree polynomial equation, typically in the form ax^2 + bx + c = 0, where ‘a,’ ‘b,’ and ‘c’ are constants.
Q2: How can I solve a quadratic equation?
A2: You can solve a quadratic equation using methods like factoring or the quadratic formula, depending on its complexity.
Q3: What is the discriminant in a quadratic equation?
A3: The discriminant is the value inside the square root in the quadratic formula and determines the number and nature of the solutions.
Q4: What are the real-world applications of quadratic equations?
A4: Quadratic equations are used in physics, engineering, economics, and various other fields to model and solve real-world problems.
Q5: Can all quadratic equations be solved using the quadratic formula?
A5: Yes, the quadratic formula can be used to solve any quadratic equation, whether it’s easily factorable or not.
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