Solve The Following Quadratic Equation For All Values Of $x$ In Simplest Form:$4(x+1)^2 - 7 = 45$

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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, $4(x+1)^2 - 7 = 45$, for all values of $x$ in simplest form. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this case, the equation is $4(x+1)^2 - 7 = 45$.

Step 1: Simplify the Equation

To simplify the equation, we need to isolate the quadratic term. We can start by adding $7$ to both sides of the equation, which gives us:

4(x+1)2=45+74(x+1)^2 = 45 + 7

4(x+1)2=524(x+1)^2 = 52

Step 2: Divide Both Sides by 4

Next, we need to divide both sides of the equation by $4$ to isolate the quadratic term:

(x+1)2=524(x+1)^2 = \frac{52}{4}

(x+1)2=13(x+1)^2 = 13

Step 3: Take the Square Root of Both Sides

Now, we need to take the square root of both sides of the equation to solve for $x+1$:

x+1=Β±13x+1 = \pm \sqrt{13}

Step 4: Simplify the Square Root

The square root of $13$ can be simplified as:

13=9+4\sqrt{13} = \sqrt{9 + 4}

13=9+4\sqrt{13} = \sqrt{9} + \sqrt{4}

13=3+2\sqrt{13} = 3 + 2

13=5\sqrt{13} = 5

However, we also need to consider the negative square root:

βˆ’13=βˆ’3βˆ’2-\sqrt{13} = -3 - 2

βˆ’13=βˆ’5-\sqrt{13} = -5

Step 5: Solve for x

Now that we have the simplified square root, we can solve for $x$ by subtracting $1$ from both sides of the equation:

x=βˆ’1+5x = -1 + 5

x=4x = 4

or

x=βˆ’1βˆ’5x = -1 - 5

x=βˆ’6x = -6

Conclusion

In this article, we solved the quadratic equation $4(x+1)^2 - 7 = 45$ for all values of $x$ in simplest form. We broke down the solution into manageable steps, making it easy to follow and understand. By simplifying the equation, dividing both sides by $4$, taking the square root of both sides, and solving for $x$, we arrived at the final solutions: $x = 4$ and $x = -6$.

Tips and Tricks

When solving quadratic equations, it's essential to remember the following tips and tricks:

  • Always simplify the equation before solving it.
  • Use the quadratic formula when necessary.
  • Check your solutions by plugging them back into the original equation.
  • Be careful when taking the square root of both sides of the equation.

By following these tips and tricks, you'll become a pro at solving quadratic equations in no time!

Real-World Applications

Quadratic equations have numerous 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.

Introduction

In our previous article, we solved the quadratic equation $4(x+1)^2 - 7 = 45$ for all values of $x$ in simplest form. In this article, we will answer some of the most frequently asked questions about quadratic equations.

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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, 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: If the quadratic expression can be factored into the product of two binomials, you can solve for $x$ by setting each factor equal to zero.
  • Quadratic formula: The quadratic formula is a formula that can be used to solve any quadratic equation. The formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  • Graphing: You can also solve a quadratic equation by graphing the related function and finding the $x$-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve any quadratic equation. The formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared term, while a linear equation does not.

Q: Can I solve a quadratic equation with a negative coefficient?

A: Yes, you can solve a quadratic equation with a negative coefficient. The process is the same as solving a quadratic equation with a positive coefficient.

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original equation and simplify. If the equation is true, then the solution is correct.

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

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

  • Not simplifying the equation before solving it.
  • Not using the quadratic formula when necessary.
  • Not checking your solutions.
  • Not being careful when taking the square root of both sides of the equation.

Conclusion

In this article, we answered some of the most frequently asked questions about quadratic equations. We covered topics such as the definition of a quadratic equation, how to solve a quadratic equation, and common mistakes to avoid. By following the tips and tricks outlined in this article, you'll become a pro at solving quadratic equations in no time!

Additional Resources

For more information on quadratic equations, check out the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations

Practice Problems

Try solving the following quadratic equations:

  • x2+5x+6=0x^2 + 5x + 6 = 0

  • x2βˆ’3xβˆ’4=0x^2 - 3x - 4 = 0

  • x2+2xβˆ’15=0x^2 + 2x - 15 = 0

By practicing these problems, you'll become more confident and proficient in solving quadratic equations.