Solve The Following Quadratic Equation For All Values Of $x$ In Simplest Form.$4(3x - 9)^2 - 9 = 7$\$x =$[/tex\] $\square$

by ADMIN 129 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, step by step, to understand the process and arrive at the solution.

The Quadratic Equation

The given quadratic equation is:

4(3x−9)2−9=74(3x - 9)^2 - 9 = 7

Our goal is to solve for all values of xx in the simplest form.

Step 1: Expand the Equation

To begin solving the equation, we need to expand the squared term using the formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2. In this case, a=3xa = 3x and b=9b = 9.

4(3x−9)2−9=74(3x - 9)^2 - 9 = 7

4(9x2−18x+81)−9=74(9x^2 - 18x + 81) - 9 = 7

Step 2: Distribute the Coefficient

Now, we need to distribute the coefficient 44 to the terms inside the parentheses.

36x2−72x+324−9=736x^2 - 72x + 324 - 9 = 7

Step 3: Combine Like Terms

Next, we combine like terms to simplify the equation.

36x2−72x+315=736x^2 - 72x + 315 = 7

Step 4: Move All Terms to One Side

To solve for xx, we need to move all terms to one side of the equation.

36x2−72x+308=036x^2 - 72x + 308 = 0

Step 5: Factor the Quadratic Equation

Unfortunately, this quadratic equation does not factor easily. Therefore, we will use the quadratic formula to solve for xx.

The Quadratic Formula

The quadratic formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a=36a = 36, b=−72b = -72, and c=308c = 308.

Step 6: Plug in the Values

Now, we plug in the values into the quadratic formula.

x=−(−72)±(−72)2−4(36)(308)2(36)x = \frac{-(-72) \pm \sqrt{(-72)^2 - 4(36)(308)}}{2(36)}

Step 7: Simplify the Expression

Next, we simplify the expression inside the square root.

x=72±5184−4420872x = \frac{72 \pm \sqrt{5184 - 44208}}{72}

x=72±−3902472x = \frac{72 \pm \sqrt{-39024}}{72}

Step 8: Simplify the Square Root

Since the square root of a negative number is not a real number, we can conclude that this quadratic equation has no real solutions.

Conclusion

In this article, we solved a quadratic equation step by step, using the quadratic formula to arrive at the solution. Unfortunately, the equation had no real solutions. However, this process can be applied to any quadratic equation to find the solutions.

Tips and Tricks

  • When solving quadratic equations, always start by expanding the squared term.
  • Use the quadratic formula when the equation does not factor easily.
  • Be careful when simplifying the expression inside the square root.
  • Remember that the square root of a negative number is not a real number.

Common Quadratic Equations

Here are some common quadratic equations that can be solved using the quadratic formula:

  • x2+5x+6=0x^2 + 5x + 6 = 0
  • x2−7x+12=0x^2 - 7x + 12 = 0
  • x2+2x−15=0x^2 + 2x - 15 = 0

Real-World Applications

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 economic systems and make predictions about future trends.

Conclusion

Frequently Asked Questions

Q: What is a quadratic equation?

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

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when the quadratic equation does not factor easily or when you are not sure how to factor it.

Q: What is the difference between a quadratic equation and a linear equation?

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (usually x) is one. A quadratic equation, on the other hand, is a polynomial equation of degree two.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula will always give you two possible values for x.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This will happen when the discriminant (b^2 - 4ac) is negative.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It is b^2 - 4ac.

Q: How do I determine the number of solutions a quadratic equation has?

A: You can determine the number of solutions a quadratic equation has by looking at the discriminant. If the discriminant is positive, the equation has two solutions. If the discriminant is zero, the equation has one solution. If the discriminant is negative, the equation has no solutions.

Q: Can a quadratic equation be used to model real-world problems?

A: Yes, quadratic equations can be used to model real-world problems. For example, the motion of an object under the influence of gravity can be modeled using a quadratic equation.

Q: What are some common applications of quadratic equations?

A: Some common applications of quadratic equations include:

  • 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 economic systems and make predictions about future trends.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to solve the equation.

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

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

  • Not following the order of operations
  • Not simplifying the expression inside the square root
  • Not checking the discriminant to determine the number of solutions

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics and have many real-world applications. By understanding how to solve quadratic equations, you can model and analyze a wide range of problems. Remember to always follow the order of operations, simplify the expression inside the square root, and check the discriminant to determine the number of solutions.