Which Of The Following Are Solutions To The Equation Below? Check All That Apply. X 2 + 5 X − 8 = 4 X + 4 X^2 + 5x - 8 = 4x + 4 X 2 + 5 X − 8 = 4 X + 4 A. 4 B. -4 C. -2 D. -3 E. 3 F. 5

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 explore the solutions to a given quadratic equation and provide a step-by-step guide on how to solve it.

The Equation

The given equation is:

$x^2 + 5x - 8 = 4x + 4$

Our goal is to find the solutions to this equation, which are the values of $x$ that satisfy the equation.

Step 1: Rearrange the Equation

To solve the equation, we need to isolate the quadratic term on one side of the equation. We can do this by subtracting $4x$ from both sides of the equation:

$x^2 + 5x - 8 - 4x = 4x + 4 - 4x$

This simplifies to:

$x^2 + x - 8 = 4$

Step 2: Move the Constant Term to the Right Side

Next, we need to move the constant term to the right side of the equation. We can do this by subtracting $4$ from both sides of the equation:

$x^2 + x - 8 - 4 = 4 - 4$

This simplifies to:

$x^2 + x - 12 = 0$

Step 3: Factor the Quadratic Expression

Now, we need to factor the quadratic expression on the left side of the equation. We can do this by finding two numbers whose product is $-12$ and whose sum is $1$. These numbers are $-3$ and $4$, so we can factor the quadratic expression as:

$(x - 3)(x + 4) = 0$

Step 4: Solve for $x$

Now that we have factored the quadratic expression, we can solve for $x$ by setting each factor equal to zero:

$x - 3 = 0$ or $x + 4 = 0$

Solving for $x$ in each equation, we get:

$x = 3$ or $x = -4$

Conclusion

In conclusion, the solutions to the given quadratic equation are $x = 3$ and $x = -4$. These values of $x$ satisfy the equation and are the correct solutions.

Checking the Solutions

To verify our solutions, we can plug each value of $x$ back into the original equation and check if it is true. If it is true, then we have found the correct solutions.

For $x = 3$, we have:

$(3)^2 + 5(3) - 8 = 9 + 15 - 8 = 16$

And for $x = -4$, we have:

$(-4)^2 + 5(-4) - 8 = 16 - 20 - 8 = -12$

Since both values of $x$ satisfy the equation, we can conclude that the solutions to the given quadratic equation are indeed $x = 3$ and $x = -4$.

Which of the Following are Solutions to the Equation?

Based on our analysis, we can conclude that the following are solutions to the equation:

• A. 4: Incorrect, since $x = 4$ does not satisfy the equation.
• B. -4: Correct, since $x = -4$ satisfies the equation.
• C. -2: Incorrect, since $x = -2$ does not satisfy the equation.
• D. -3: Correct, since $x = -3$ satisfies the equation.
• E. 3: Correct, since $x = 3$ satisfies the equation.
• F. 5: Incorrect, since $x = 5$ does not satisfy the equation.

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Introduction

In our previous article, we explored the solutions to a given quadratic equation and provided a step-by-step guide on how to solve it. In this article, we will answer some frequently asked questions about quadratic equations and their solutions.

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 (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: To solve a quadratic equation, you can use the following steps:

1. Rearrange the equation to isolate the quadratic term on one side.
2. Move the constant term to the right side of the equation.
3. Factor the quadratic expression, if possible.
4. Solve for $x$ by setting each factor equal to zero.

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of $x$ that satisfy the equation. These values can be found by solving for $x$ using the steps outlined above.

Q: How do I know if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you can use the discriminant, which is the expression under the square root in the quadratic formula:

$\Delta = b^2 - 4ac$

If $\Delta > 0$, then the equation has two real solutions. If $\Delta = 0$, then the equation has one real solution. If $\Delta < 0$, then the equation has two complex solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the solutions to a quadratic equation. It is given by:

$x = \frac{-b \pm \sqrt{\Delta}}{2a}$

where $\Delta$ is the discriminant.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, you can use the quadratic formula to solve any quadratic equation. However, if the equation can be factored easily, it may be more efficient to use factoring to solve it.

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

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

• Not rearranging the equation to isolate the quadratic term.
• Not moving the constant term to the right side of the equation.
• Not factoring the quadratic expression, if possible.
• Not checking the solutions to see if they satisfy the original equation.

Conclusion

In conclusion, solving quadratic equations can be a challenging task, but with the right tools and techniques, it can be done efficiently and effectively. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving quadratic equations and apply this skill to a wide range of mathematical and real-world problems.

Frequently Asked Questions

• 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.
• Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions? A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions.
• Q: How do I know if a quadratic equation has a real or complex solution? A: You can use the discriminant to determine if a quadratic equation has a real or complex solution.
• Q: Can I use factoring to solve a quadratic equation with complex solutions? A: No, you cannot use factoring to solve a quadratic equation with complex solutions.

Additional Resources

• Quadratic Equation Solver: A online tool that can be used to solve quadratic equations.
• Quadratic Formula Calculator: A online tool that can be used to calculate the solutions to a quadratic equation using the quadratic formula.
• Quadratic Equations Tutorial: A tutorial that provides a step-by-step guide on how to solve quadratic equations.