Use The Quadratic Formula To Solve The Equation. Enter Your Answers As A Comma-separated List.$5x^2 + X = -5$x =$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and it is essential to understand how to use it effectively. In this article, we will explore how to use the quadratic formula to solve the equation $5x^2 + x = -5$.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

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

Understanding the Quadratic Formula

To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the quadratic equation. In the equation $5x^2 + x = -5$, we can rewrite it as $5x^2 + x + 5 = 0$. Comparing this with the standard form of a quadratic equation, we can see that $a = 5$, $b = 1$, and $c = 5$.

Applying the Quadratic Formula

Now that we have identified the values of $a$, $b$, and $c$, we can plug them into the quadratic formula:

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

Substituting the values of $a$, $b$, and $c$, we get:

$x = \frac{-1 \pm \sqrt{1^2 - 4(5)(5)}}{2(5)}$

Simplifying the expression, we get:

$x = \frac{-1 \pm \sqrt{1 - 100}}{10}$

$x = \frac{-1 \pm \sqrt{-99}}{10}$

Simplifying the Square Root

The square root of a negative number is an imaginary number, which can be represented as $i\sqrt{99}$. Therefore, we can simplify the expression as:

$x = \frac{-1 \pm i\sqrt{99}}{10}$

Simplifying the Imaginary Number

We can simplify the imaginary number $i\sqrt{99}$ by factoring out the square root of 9:

$i\sqrt{99} = i\sqrt{9 \cdot 11} = 3i\sqrt{11}$

Therefore, we can simplify the expression as:

$x = \frac{-1 \pm 3i\sqrt{11}}{10}$

Conclusion

In this article, we have used the quadratic formula to solve the equation $5x^2 + x = -5$. We have identified the values of $a$, $b$, and $c$, and plugged them into the quadratic formula. We have also simplified the expression to obtain the solutions to the equation. The solutions are complex numbers, which can be represented as $x = \frac{-1 \pm 3i\sqrt{11}}{10}$.

Final Answer

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Introduction

The quadratic formula is a powerful tool for solving quadratic equations, but it can be a bit tricky to understand and apply. In this article, we will answer some common questions about the quadratic formula and provide examples to help illustrate the concepts.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, you can plug these values into the formula and simplify the expression to obtain the solutions.

Q: What if the quadratic equation has no real solutions?

A: If the quadratic equation has no real solutions, it means that the expression under the square root is negative. In this case, the solutions will be complex numbers, which can be represented as $x = \frac{-b \pm i\sqrt{b^2 - 4ac}}{2a}$.

Q: How do I simplify complex solutions?

A: To simplify complex solutions, you can use the fact that $i^2 = -1$. This means that you can multiply the numerator and denominator of the complex solution by $i$ to eliminate the square root.

Q: What if the quadratic equation has repeated solutions?

A: If the quadratic equation has repeated solutions, it means that the expression under the square root is zero. In this case, the solutions will be repeated, and you can simplify the expression to obtain the repeated solution.

Q: Can I use the quadratic formula to solve quadratic inequalities?

A: No, the quadratic formula is only used to solve quadratic equations, not quadratic inequalities. To solve quadratic inequalities, you need to use a different method, such as factoring or graphing.

Q: Are there any special cases where the quadratic formula does not work?

A: Yes, there are a few special cases where the quadratic formula does not work. For example, if the quadratic equation has a variable in the denominator, the quadratic formula will not work. In this case, you need to use a different method to solve the equation.

Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you need to be careful when simplifying the expression, as complex coefficients can lead to complex solutions.

Conclusion

In this article, we have answered some common questions about the quadratic formula and provided examples to help illustrate the concepts. We hope that this article has been helpful in understanding the quadratic formula and how to use it to solve quadratic equations.

Final Tips

• Make sure to identify the values of $a$, $b$, and $c$ in the quadratic equation before using the quadratic formula.
• Simplify the expression carefully to avoid errors.
• Use the quadratic formula to solve quadratic equations, not quadratic inequalities.
• Be careful when simplifying complex solutions, as complex coefficients can lead to complex solutions.

Common Quadratic Formula Mistakes

• Forgetting to identify the values of $a$, $b$, and $c$ in the quadratic equation.
• Simplifying the expression incorrectly.
• Using the quadratic formula to solve quadratic inequalities.
• Not being careful when simplifying complex solutions.

Final Answer

The final answer is: Use the quadratic formula to solve quadratic equations, and be careful when simplifying complex solutions.