Which Shows The Correct Substitution Of The Values \[$ A, B, \$\] And \[$ C \$\] From The Equation \[$ 1 = -2x + 3x^2 + 1 \$\] Into The Quadratic Formula?Quadratic Formula: $\[ X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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Understanding the Quadratic Formula

The quadratic formula is a fundamental concept in algebra, 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} $}$. In this equation, { a $}$, { b $}$, and { c $}$ are the coefficients of the quadratic equation.

Identifying the Coefficients

To apply the quadratic formula, we need to identify the values of { a $}$, { b $}$, and { c $}$ from the given equation { 1 = -2x + 3x^2 + 1 $}$. By comparing the given equation with the standard form of a quadratic equation { ax^2 + bx + c = 0 $}$, we can see that:

  • { a = 3 $}$, which is the coefficient of the { x^2 $}$ term.
  • { b = -2 $}$, which is the coefficient of the { x $}$ term.
  • { c = 1 $}$, which is the constant term.

Substituting the Values 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{-(-2) \pm \sqrt{(-2)^2 - 4(3)(1)}}{2(3)} $}$

Simplifying the Expression

To simplify the expression, we can start by evaluating the terms inside the square root:

{ (-2)^2 = 4 $} {$ 4(3)(1) = 12 $}$

Substituting these values back into the expression, we get:

{ x = \frac{2 \pm \sqrt{4 - 12}}{6} $}$

Continuing to Simplify

Next, we can simplify the expression inside the square root:

{ 4 - 12 = -8 $}$

Since the square root of a negative number is not a real number, we need to consider the two possible cases:

  • { \sqrt{-8} $}$ is not a real number, so we need to use the imaginary unit { i $}$ to represent it.
  • { \sqrt{-8} $}$ can be simplified as { \sqrt{-8} = \sqrt{-1} \cdot \sqrt{8} = 2i\sqrt{2} $}$.

Simplifying the Expression Further

Using the simplified expression for the square root, we can rewrite the quadratic formula as:

{ x = \frac{2 \pm 2i\sqrt{2}}{6} $}$

Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 2:

{ x = \frac{1 \pm i\sqrt{2}}{3} $}$

Conclusion

In conclusion, the correct substitution of the values { a, b, $}$ and { c $}$ from the equation { 1 = -2x + 3x^2 + 1 $}$ into the quadratic formula is { x = \frac{1 \pm i\sqrt{2}}{3} $}$. This result shows that the quadratic equation has two complex solutions.

Discussion

The quadratic formula is a powerful tool for solving quadratic equations, but it can be challenging to apply in certain cases. In this example, we saw how to substitute the values of { a, b, $}$ and { c $}$ from the given equation into the quadratic formula and simplify the resulting expression. The result shows that the quadratic equation has two complex solutions, which can be represented using the imaginary unit { i $}$.

Real-World Applications

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

  • Physics: The quadratic formula is used to describe the motion of objects under the influence of gravity or other forces.
  • Engineering: The quadratic formula is used to design and optimize systems, such as bridges and buildings.
  • Computer Science: The quadratic formula is used in algorithms for solving problems, such as finding the shortest path between two points.

Conclusion

In conclusion, the quadratic formula is a fundamental concept in algebra that has many real-world applications. By understanding how to substitute the values of { a, b, $}$ and { c $}$ from a given equation into the quadratic formula, we can solve quadratic equations and apply the results to a wide range of problems.

Final Thoughts

The quadratic formula is a powerful tool for solving quadratic equations, but it requires careful attention to detail and a deep understanding of the underlying mathematics. By mastering the quadratic formula, we can solve a wide range of problems and apply the results to real-world situations.

References

  • "Algebra" by Michael Artin
  • "Quadratic Equations" by Paul Halmos
  • "The Quadratic Formula" by Wolfram MathWorld

Further Reading

  • "Quadratic Equations and Functions" by Khan Academy
  • "The Quadratic Formula" by Math Is Fun
  • "Quadratic Equations" by Purplemath

Understanding the Quadratic Formula

The quadratic formula is a fundamental concept in algebra, 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} $}$. In this article, we will answer some of the most frequently asked questions about the quadratic formula.

Q: What is the Quadratic Formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form { ax^2 + bx + c = 0 $}$. It 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 $}$ from the given equation. Then, substitute these values into the quadratic formula and simplify the resulting expression.

Q: What is the Difference Between the Quadratic Formula and Factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions.

Q: Can the Quadratic Formula Be Used to Solve All Quadratic Equations?

A: Yes, the quadratic formula can be used to solve all quadratic equations of the form { ax^2 + bx + c = 0 $}$. However, it may not always be the most efficient method, especially for simple equations that can be factored easily.

Q: What is the Significance of the Imaginary Unit { i $}$ in the Quadratic Formula?

A: The imaginary unit { i $}$ is used to represent the square root of -1. In the quadratic formula, it is used to simplify the expression inside the square root.

Q: Can the Quadratic Formula Be Used to Solve Equations with Complex Solutions?

A: Yes, the quadratic formula can be used to solve equations with complex solutions. In these cases, the solutions will involve the imaginary unit { i $}$.

Q: How Do I Determine Whether a Quadratic Equation Has Real or Complex Solutions?

A: To determine whether a quadratic equation has real or complex solutions, you need to examine the expression inside the square root. If it is positive, the equation has real solutions. If it is negative, the equation has complex solutions.

Q: Can the Quadratic Formula Be Used to Solve Equations with Rational or Irrational Solutions?

A: Yes, the quadratic formula can be used to solve equations with rational or irrational solutions. In these cases, the solutions will be expressed as fractions or decimals.

Q: What is the Relationship Between the Quadratic Formula and the Discriminant?

A: The discriminant is the expression inside the square root in the quadratic formula, { b^2 - 4ac $}$. The sign of the discriminant determines whether the equation has real or complex solutions.

Q: Can the Quadratic Formula Be Used to Solve Equations with Multiple Solutions?

A: Yes, the quadratic formula can be used to solve equations with multiple solutions. In these cases, the solutions will be expressed as a list of values.

Q: How Do I Use the Quadratic Formula to Solve Equations with Multiple Variables?

A: To use the quadratic formula to solve equations with multiple variables, you need to identify the values of { a $}$, { b $}$, and { c $}$ from the given equation. Then, substitute these values into the quadratic formula and simplify the resulting expression.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By understanding how to use the quadratic formula, you can solve a wide range of problems and apply the results to real-world situations.

Final Thoughts

The quadratic formula is a fundamental concept in algebra that has many real-world applications. By mastering the quadratic formula, you can solve quadratic equations and apply the results to a wide range of problems.

References

  • "Algebra" by Michael Artin
  • "Quadratic Equations" by Paul Halmos
  • "The Quadratic Formula" by Wolfram MathWorld

Further Reading

  • "Quadratic Equations and Functions" by Khan Academy
  • "The Quadratic Formula" by Math Is Fun
  • "Quadratic Equations" by Purplemath