Solve \[$(x+4)^2 - 3(x+4) - 3 = 0\$\] Using Substitution.1. Let \[$u = X + 4\$\].2. Substitute \[$u\$\] Into The Equation: \[$(u)^2 - 3u - 3 = 0\$\]3. Solve The Equation For \[$u\$\].4. Once You Have The

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Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students. However, with the right approach and techniques, solving quadratic equations can be a breeze. In this article, we will explore one of the most effective methods for solving quadratic equations: substitution. We will use this method to solve the equation {(x+4)^2 - 3(x+4) - 3 = 0$}$.

Step 1: Let {u = x + 4$}$

The first step in solving the equation using substitution is to introduce a new variable, {u$}$, which is equal to {x + 4$}$. This allows us to rewrite the original equation in terms of {u$}$, making it easier to solve.

Step 2: Substitute {u$}$ into the equation

Now that we have introduced the new variable {u$}$, we can substitute it into the original equation. The equation becomes:

{(u)^2 - 3u - 3 = 0$}$

This is a quadratic equation in terms of {u$}$, and we can now use standard methods to solve it.

Step 3: Solve the equation for {u$}$

To solve the equation {(u)^2 - 3u - 3 = 0$}$, we can use the quadratic formula:

{u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$}$

In this case, {a = 1$}$, {b = -3$}$, and {c = -3$}$. Plugging these values into the quadratic formula, we get:

{u = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-3)}}{2(1)}$}$

Simplifying the expression, we get:

{u = \frac{3 \pm \sqrt{9 + 12}}{2}$}$

{u = \frac{3 \pm \sqrt{21}}{2}$}$

Therefore, the solutions to the equation {(u)^2 - 3u - 3 = 0$}$ are:

{u = \frac{3 + \sqrt{21}}{2}$}$ and {u = \frac{3 - \sqrt{21}}{2}$}$

Step 4: Find the solutions for {x$}$

Now that we have found the solutions for {u$}$, we can substitute back to find the solutions for {x$}$. Recall that {u = x + 4$}$, so we can rewrite the solutions as:

{x + 4 = \frac{3 + \sqrt{21}}{2}$}$ and {x + 4 = \frac{3 - \sqrt{21}}{2}$}$

Subtracting 4 from both sides of each equation, we get:

{x = \frac{3 + \sqrt{21}}{2} - 4$}$ and {x = \frac{3 - \sqrt{21}}{2} - 4$}$

Simplifying the expressions, we get:

{x = \frac{3 + \sqrt{21} - 8}{2}$}$ and {x = \frac{3 - \sqrt{21} - 8}{2}$}$

{x = \frac{-5 + \sqrt{21}}{2}$}$ and {x = \frac{-5 - \sqrt{21}}{2}$}$

Therefore, the solutions to the original equation {(x+4)^2 - 3(x+4) - 3 = 0$}$ are {x = \frac{-5 + \sqrt{21}}{2}$}$ and {x = \frac{-5 - \sqrt{21}}{2}$}$.

Conclusion

In this article, we used the substitution method to solve the quadratic equation {(x+4)^2 - 3(x+4) - 3 = 0$}$. We introduced a new variable {u$}$ and substituted it into the original equation, making it easier to solve. We then used the quadratic formula to find the solutions for {u$}$ and finally substituted back to find the solutions for {x$}$. This method is a powerful tool for solving quadratic equations and can be applied to a wide range of problems.

Frequently Asked Questions

  • Q: What is the substitution method? A: The substitution method is a technique used to solve quadratic equations by introducing a new variable and substituting it into the original equation.
  • Q: How do I know when to use the substitution method? A: You can use the substitution method when the quadratic equation can be factored or when the equation can be rewritten in a simpler form using a new variable.
  • Q: What are the steps involved in solving a quadratic equation using substitution? A: The steps involved in solving a quadratic equation using substitution are:
    1. Introduce a new variable and substitute it into the original equation.
    2. Solve the equation for the new variable.
    3. Substitute back to find the solutions for the original variable.

Further Reading

  • Quadratic Formula: A comprehensive guide to the quadratic formula and its applications.
  • Factoring Quadratic Equations: A step-by-step guide to factoring quadratic equations.
  • Solving Quadratic Equations: A collection of resources and tutorials on solving quadratic equations.

References

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students. However, with the right approach and techniques, solving quadratic equations can be a breeze. In this article, we will answer some of the most frequently asked questions about quadratic equations and provide a comprehensive guide to solving them.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two. It is typically written in the form {ax^2 + bx + c = 0$}$, where {a$}$, {b$}$, and {c$}$ are constants, and {x$}$ is the variable.

Q: What are the different methods for solving quadratic equations?

A: There are several methods for solving quadratic equations, including:

  • Factoring: This involves expressing the quadratic equation as a product of two binomials.
  • Quadratic Formula: This involves using the quadratic formula to find the solutions to the equation.
  • Substitution: This involves introducing a new variable and substituting it into the original equation.
  • Graphing: This involves graphing the quadratic equation on a coordinate plane and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the solutions to a quadratic equation. It is written in the form:

{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$}$

where {a$}$, {b$}$, and {c$}$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to plug in the values of {a$}$, {b$}$, and {c$}$ into the formula and simplify. The formula will give you two solutions, which are the x-coordinates of the x-intercepts of the quadratic equation.

Q: What is the difference between the quadratic formula and the factoring method?

A: The quadratic formula and the factoring method are two different methods for solving quadratic equations. The quadratic formula is a general method that can be used to solve any quadratic equation, while the factoring method is a specific method that can be used to solve quadratic equations that can be factored.

Q: Can I use the quadratic formula to solve quadratic equations that cannot be factored?

A: Yes, you can use the quadratic formula to solve quadratic equations that cannot be factored. The quadratic formula is a general method that can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

  • Not checking if the equation can be factored before using the quadratic formula.
  • Not simplifying the expression under the square root.
  • Not checking if the solutions are real or complex.
  • Not checking if the solutions are rational or irrational.

Q: How do I check if the solutions to a quadratic equation are real or complex?

A: To check if the solutions to a quadratic equation are real or complex, you need to check if the expression under the square root is positive or negative. If the expression is positive, the solutions are real. If the expression is negative, the solutions are complex.

Q: How do I check if the solutions to a quadratic equation are rational or irrational?

A: To check if the solutions to a quadratic equation are rational or irrational, you need to check if the solutions can be expressed as a ratio of integers or if they are non-terminating decimals.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand curves.
  • Computer Science: Quadratic equations are used to solve problems in computer science, such as finding the shortest path in a graph.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students. However, with the right approach and techniques, solving quadratic equations can be a breeze. In this article, we have answered some of the most frequently asked questions about quadratic equations and provided a comprehensive guide to solving them. We hope that this article has been helpful in understanding quadratic equations and their applications.

Further Reading

  • Quadratic Formula: A comprehensive guide to the quadratic formula and its applications.
  • Factoring Quadratic Equations: A step-by-step guide to factoring quadratic equations.
  • Solving Quadratic Equations: A collection of resources and tutorials on solving quadratic equations.

References