Using The Quadratic Formula To Solve X 2 = 5 − X X^2 = 5 - X X 2 = 5 − X , What Are The Values Of X X X ?A. − 1 ± 21 2 \frac{-1 \pm \sqrt{21}}{2} 2 − 1 ± 21 ​ ​ B. − 1 ± 19 I 2 \frac{-1 \pm \sqrt{19} I}{2} 2 − 1 ± 19 ​ I ​ C. 5 ± 21 2 \frac{5 \pm \sqrt{21}}{2} 2 5 ± 21 ​ ​ D. $\frac{1 \pm \sqrt{19}

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 how to use the quadratic formula to solve quadratic equations, with a focus on the equation $x^2 = 5 - x$. We will examine the steps involved in applying the quadratic formula, and provide a detailed solution to the given equation.

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:

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

This formula is a powerful tool for solving quadratic equations, and it is widely used in mathematics, science, and engineering.

Applying the Quadratic Formula to the Equation $x^2 = 5 - x$

To apply the quadratic formula to the equation $x^2 = 5 - x$, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting $5$ from both sides of the equation, which gives us:

$$x^2 + x - 5 = 0$$

Now that we have the equation in the standard form, we can identify the values of $a$, $b$, and $c$. In this case, $a = 1$, $b = 1$, and $c = -5$.

Substituting the Values of $a$, $b$, and $c$ into the Quadratic Formula

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

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

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

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

Simplifying the expression under the square root, we get:

$$x = \frac{-1 \pm \sqrt{1 + 20}}{2}$$

$$x = \frac{-1 \pm \sqrt{21}}{2}$$

Evaluating the Solutions

The quadratic formula gives us two possible solutions for the equation $x^2 = 5 - x$. These solutions are given by:

$$x = \frac{-1 + \sqrt{21}}{2}$$

$$x = \frac{-1 - \sqrt{21}}{2}$$

These solutions can be evaluated using a calculator or by hand.

Conclusion

In this article, we have used the quadratic formula to solve the equation $x^2 = 5 - x$. We have identified the values of $a$, $b$, and $c$, and substituted them into the quadratic formula. We have also evaluated the solutions to the equation. The quadratic formula is a powerful tool for solving quadratic equations, and it is widely used in mathematics, science, and engineering.

Discussion

The quadratic formula is a fundamental concept in mathematics, and it is used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

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

This formula is a powerful tool for solving quadratic equations, and it is widely used in mathematics, science, and engineering.

Common Mistakes

When using the quadratic formula, there are several common mistakes that can be made. These include:

• Incorrectly identifying the values of $a$, $b$, and $c$: This can lead to incorrect solutions to the equation.
• Incorrectly substituting the values of $a$, $b$, and $c$ into the quadratic formula: This can also lead to incorrect solutions to the equation.
• Not evaluating the solutions to the equation: This can lead to incorrect solutions to the equation.

Tips and Tricks

When using the quadratic formula, there are several tips and tricks that can be used to ensure that the solutions to the equation are correct. These include:

• Double-checking the values of $a$, $b$, and $c$: This can help to ensure that the solutions to the equation are correct.
• Carefully substituting the values of $a$, $b$, and $c$ into the quadratic formula: This can help to ensure that the solutions to the equation are correct.
• Evaluating the solutions to the equation: This can help to ensure that the solutions to the equation are correct.

Conclusion

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Frequently Asked Questions

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 to solve a quadratic equation?

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

Q: What are the values of $a$, $b$, and $c$ in the quadratic formula?

A: The values of $a$, $b$, and $c$ in the quadratic formula are the coefficients of the quadratic equation. In the equation $ax^2 + bx + c = 0$, $a$ is the coefficient of the $x^2$ term, $b$ is the coefficient of the $x$ term, and $c$ is the constant term.

Q: How do I evaluate the solutions to a quadratic equation using the quadratic formula?

A: To evaluate the solutions to a quadratic equation using the quadratic formula, you need to simplify the expression under the square root and then solve for $x$. You can use a calculator or simplify the expression by hand.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Incorrectly identifying the values of $a$, $b$, and $c$: This can lead to incorrect solutions to the equation.
• Incorrectly substituting the values of $a$, $b$, and $c$ into the quadratic formula: This can also lead to incorrect solutions to the equation.
• Not evaluating the solutions to the equation: This can lead to incorrect solutions to the equation.

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

A: To use the quadratic formula to solve a quadratic equation with complex solutions, you need to simplify the expression under the square root and then solve for $x$. You can use a calculator or simplify the expression by hand.

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

A: The quadratic formula has many real-world applications, including:

• Physics: The quadratic formula is used to solve problems involving motion, such as the trajectory of a projectile.
• Engineering: The quadratic formula is used to solve problems involving the design of structures, such as bridges and buildings.
• Computer Science: The quadratic formula is used to solve problems involving algorithms and data structures.

Q: How do I choose between the quadratic formula and other methods for solving quadratic equations?

A: When choosing between the quadratic formula and other methods for solving quadratic equations, you should consider the following factors:

• Complexity of the equation: If the equation is simple, you may be able to solve it using factoring or other methods. If the equation is complex, the quadratic formula may be the best choice.
• Number of solutions: If the equation has two distinct solutions, the quadratic formula may be the best choice. If the equation has only one solution, other methods may be more efficient.
• Accuracy: If you need to find the solutions to the equation with high accuracy, the quadratic formula may be the best choice.

Conclusion

In this article, we have answered some frequently asked questions about the quadratic formula. We have discussed how to use the quadratic formula to solve quadratic equations, how to evaluate the solutions to a quadratic equation, and how to choose between the quadratic formula and other methods for solving quadratic equations. We hope that this article has been helpful in answering your questions about the quadratic formula.