Even the most experienced mathematicians encounter issues with solving more complex equations in the quadratic formula, such as the given equation of (X – 4)2 – (X – 4) – 6 = 0. To solve this equation correctly, you’ll need to understand what a quadratic equation is and then have a clear understanding of each of the steps in the process. This article will explain the basics of quadratic equations and provide an in-depth analysis of the given equation, giving you everything you need to know to find the right solution.

Definition of Quadratic Equations

In mathematics, a quadratic equation is an equation of the form ax^2 + bx + c = 0, where x is a variable and a, b, and c are constants. These equations are usually used to find the roots of a quadratic function, which are the x-intercepts of the graph of the function. Quadratic equations can be easily recognized because the highest power of x is 2.

Quadratic equations can be solved using a variety of methods, including factoring, completing the square, and using the quadratic formula. Each method has its own advantages and disadvantages, so it is important to understand the different methods and when to use them. Additionally, quadratic equations can be used to model real-world problems, such as projectile motion and the motion of a pendulum.

How to Find an Equivalent Quadratic Equation

To find an equivalent quadratic equation for (X – 4)2 – (X – 4) – 6 = 0, first use the distributive law which states that (X – 4)(X – 4) = X2 – 8X + 16. Then combine like terms to get X2 – 8X + 16 – X + 4 – 6 = 0. Rearranging gives a standard quadratic form of X2 – 7X – 2 = 0. This is the equivalent quadratic equation of (X – 4)2 – (X– 4) – 6 = 0.

Once you have the equivalent quadratic equation, you can use the quadratic formula to solve for the two solutions. The quadratic formula is x = [-b ± √(b2 – 4ac)]/2a. In this case, a = 1, b = -7, and c = -2. Plugging these values into the formula gives x = [7 ± √(49 + 8)]/2, which simplifies to x = 4 ± √13/2.

The Process of Solving the Given Equation

To solve this equation, start by factorising it using the quadratic formula as X2 – 7X – 2 = (X – 2)(X + 1) = 0. This factorisation gives you two equations: X – 2 = 0 and X + 1 = 0. Solving each equation yields X = 2 and X = -1. Therefore there are two solutions to the given equation: X = 2 and X = -1.

It is important to note that the two solutions are not necessarily the only solutions to the equation. It is possible that there are other solutions that are not real numbers, such as complex numbers. To find these solutions, you can use the quadratic formula to solve the equation. This will give you two solutions, one of which is a real number and the other a complex number.

Overview of the Solution Steps

To solve (X – 4)2 – (X – 4) – 6 = 0, use the following steps to find the right solution:

  • Convert the equation (X – 4)2 – (X – 4) – 6 = 0 into a standard quadratic equation: X2 – 7X – 2 = 0.
  • Factorise the equation using the quadratic formula: (X – 2)(X + 1) = 0.
  • Solve the two equations, X – 2 = 0 and X + 1 = 0, to get X = 2 and X = -1.
  • The two solutions for the given equation are X = 2 and X = -1.

Examples of Other Similar Equations

Another example of a quadratic equation similar to (X – 4)2 – (X– 4)– 6 = 0 is (X – 5)(X + 3) = 0. To solve this equation, first factorise it and then solve the equations generated. This will yield two solutions: X = 5 and X = -3.

Common Mistakes to Avoid when Solving Quadratic Equations

When solving quadratic equations, you should always double-check your results to make sure they are correct. Another mistake that people often make is forgetting to use the quadratic formula to factorise the equation before trying to solve it. It is also important to remember that there may be multiple solutions for a single equation, so you should check them all carefully.

Conclusion

Solving quadratic equations may seem daunting at first, but with a little bit of knowledge and practice, anyone can master this type of problem. This article provides a detailed explanation of how to solve (X – 4)2 – (X– 4)– 6 = 0, from understanding what a quadratic equation is to finding the solutions of this specific one. Remember to always double-check your results and you’ll be able to solve any quadratic problem in no time!