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

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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 explore the quadratic formula and its application to solve quadratic equations. We will use the equation $x^2=5-x$ as a case study to demonstrate the step-by-step process of solving quadratic equations using the quadratic formula.

What is the Quadratic Formula?

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

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Applying the Quadratic Formula to the Given Equation

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

x2+x−5=0x^2+x-5=0

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

Substituting the Coefficients into the Quadratic Formula

Now that we have the coefficients, we can substitute them into the quadratic formula:

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

Substituting $a=1$, $b=1$, and $c=-5$ into the formula, we get:

x=−1±12−4(1)(−5)2(1)x=\frac{-1 \pm \sqrt{1^2-4(1)(-5)}}{2(1)}

Simplifying the Expression

Now that we have the expression, we can simplify it by evaluating the expression inside the square root:

x=−1±1+202x=\frac{-1 \pm \sqrt{1+20}}{2}

x=−1±212x=\frac{-1 \pm \sqrt{21}}{2}

Conclusion

In this article, we used the quadratic formula to solve the quadratic equation $x^2=5-x$. We identified the coefficients $a$, $b$, and $c$, substituted them into the quadratic formula, and simplified the expression to get the solutions $x=\frac{-1 \pm \sqrt{21}}{2}$. This demonstrates the step-by-step process of solving quadratic equations using the quadratic formula.

Comparison of Solutions

Now that we have the solutions, let's compare them to the given options:

  • Option A: $\frac{-1 \pm \sqrt{21}}{2}$
  • Option B: $\frac{-1 \pm \sqrt{19} i}{2}$
  • Option C: $\frac{5 \pm \sqrt{21}}{2}$
  • Option D: (no solution)

Based on our calculations, we can see that the correct solution is:

  • Option A: $\frac{-1 \pm \sqrt{21}}{2}$

This is the solution we obtained using the quadratic formula.

Conclusion

In conclusion, solving quadratic equations using the quadratic formula is a straightforward process that involves identifying the coefficients, substituting them into the formula, and simplifying the expression. By following these steps, we can solve quadratic equations and obtain the solutions. In this article, we used the quadratic formula to solve the quadratic equation $x^2=5-x$ and obtained the solutions $x=\frac{-1 \pm \sqrt{21}}{2}$. This demonstrates the power and versatility of the quadratic formula in solving quadratic equations.

Final Thoughts

Solving quadratic equations is an essential skill for students and professionals alike. By mastering the quadratic formula, we can solve quadratic equations and obtain the solutions. In this article, we demonstrated the step-by-step process of solving quadratic equations using the quadratic formula and obtained the solutions $x=\frac{-1 \pm \sqrt{21}}{2}$. This is a valuable skill that can be applied to a wide range of mathematical problems and real-world applications.

Recommendations

Based on our experience with solving quadratic equations using the quadratic formula, we recommend the following:

  • Practice solving quadratic equations using the quadratic formula to develop your skills and confidence.
  • Use the quadratic formula to solve quadratic equations in different forms, such as $ax^2+bx+c=0$ and $x^2+bx+c=0$.
  • Apply the quadratic formula to solve quadratic equations in real-world applications, such as physics, engineering, and economics.

By following these recommendations, you can develop your skills and confidence in solving quadratic equations using the quadratic formula.

Glossary

  • Quadratic equation: An equation of the form $ax^2+bx+c=0$, where $a$, $b$, and $c$ are the coefficients.
  • Quadratic formula: A mathematical formula that provides the solutions to quadratic equations of the form $ax^2+bx+c=0$.
  • Coefficients: The numbers that are multiplied by the variables in a quadratic equation.
  • Solutions: The values of the variable that satisfy the quadratic equation.

By understanding these terms, you can better appreciate the quadratic formula and its application to solving quadratic equations.

References

These resources provide additional information and practice problems to help you develop your skills in solving quadratic equations using the quadratic formula.

Conclusion

In conclusion, solving quadratic equations using the quadratic formula is a powerful tool that can be applied to a wide range of mathematical problems and real-world applications. By mastering the quadratic formula, we can solve quadratic equations and obtain the solutions. In this article, we demonstrated the step-by-step process of solving quadratic equations using the quadratic formula and obtained the solutions $x=\frac{-1 \pm \sqrt{21}}{2}$. This is a valuable skill that can be applied to a wide range of mathematical problems and real-world applications.

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Introduction

In our previous article, we explored the quadratic formula and its application to solve quadratic equations. In this article, we will address some of the most frequently asked questions about the quadratic formula. Whether you are a student, teacher, or professional, this article will provide you with the answers you need to master the quadratic formula.

Q: What is the quadratic formula?

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

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

Q: How do I apply the quadratic formula to solve a quadratic equation?

A: To apply the quadratic formula, you need to identify the coefficients $a$, $b$, and $c$ in the quadratic equation. Then, substitute these coefficients into the quadratic formula and simplify the expression to get the solutions.

Q: What are the coefficients in a quadratic equation?

A: The coefficients in a quadratic equation are the numbers that are multiplied by the variables. In the quadratic equation $ax^2+bx+c=0$, the coefficients are $a$, $b$, and $c$.

Q: How do I simplify the expression in the quadratic formula?

A: To simplify the expression in the quadratic formula, you need to evaluate the expression inside the square root and then simplify the resulting expression.

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of the variable that satisfy the equation. In other words, they are the values of $x$ that make the equation true.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex solutions. In fact, the quadratic formula can be used to solve quadratic equations with complex coefficients as well.

Q: How do I determine the nature of the solutions to a quadratic equation?

A: To determine the nature of the solutions to a quadratic equation, you need to examine the discriminant $b^2-4ac$. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with rational coefficients. In fact, the quadratic formula can be used to solve quadratic equations with rational coefficients as well as irrational coefficients.

Q: How do I apply the quadratic formula to solve quadratic equations in different forms?

A: To apply the quadratic formula to solve quadratic equations in different forms, you need to rewrite the equation in the standard form $ax^2+bx+c=0$ and then substitute the coefficients into the quadratic formula.

Q: Can I use the quadratic formula to solve quadratic equations with multiple variables?

A: No, the quadratic formula is only applicable to quadratic equations with a single variable. If you have a quadratic equation with multiple variables, you need to use a different method to solve it.

Q: How do I check the solutions to a quadratic equation?

A: To check the solutions to a quadratic equation, you need to substitute the solutions back into the original equation and verify that they satisfy the equation.

Q: Can I use the quadratic formula to solve quadratic equations with non-integer coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with non-integer coefficients. In fact, the quadratic formula can be used to solve quadratic equations with non-integer coefficients as well as integer coefficients.

Conclusion

In conclusion, the quadratic formula is a powerful tool that can be used to solve quadratic equations. By mastering the quadratic formula, you can solve quadratic equations and obtain the solutions. In this article, we addressed some of the most frequently asked questions about the quadratic formula and provided you with the answers you need to master the quadratic formula.

Glossary

  • Quadratic equation: An equation of the form $ax^2+bx+c=0$, where $a$, $b$, and $c$ are the coefficients.
  • Quadratic formula: A mathematical formula that provides the solutions to quadratic equations of the form $ax^2+bx+c=0$.
  • Coefficients: The numbers that are multiplied by the variables in a quadratic equation.
  • Solutions: The values of the variable that satisfy the quadratic equation.
  • Discriminant: The expression $b^2-4ac$, which determines the nature of the solutions to a quadratic equation.

By understanding these terms, you can better appreciate the quadratic formula and its application to solving quadratic equations.

References

These resources provide additional information and practice problems to help you develop your skills in solving quadratic equations using the quadratic formula.

Conclusion

In conclusion, the quadratic formula is a powerful tool that can be used to solve quadratic equations. By mastering the quadratic formula, you can solve quadratic equations and obtain the solutions. In this article, we addressed some of the most frequently asked questions about the quadratic formula and provided you with the answers you need to master the quadratic formula.