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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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 apply it correctly. In this article, we will explore the correct substitution of values from a given quadratic equation into the quadratic formula.

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 as follows:

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

Given Quadratic Equation

The given quadratic equation is:

1=−2x+3x2+11 = -2x + 3x^2 + 1

Substituting Values into the Quadratic Formula

To solve the given quadratic equation using the quadratic formula, we need to substitute the values of a, b, and c into the formula. The values are:

  • a = 3
  • b = -2
  • c = 0

Step 1: Substitute the Values of a, b, and c into the Quadratic Formula

Substituting the values of a, b, and c into the quadratic formula, we get:

x=−(−2)±(−2)2−4(3)(0)2(3)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}

Step 2: Simplify the Expression

Simplifying the expression, we get:

x=2±4−06x = \frac{2 \pm \sqrt{4 - 0}}{6}

Step 3: Simplify the Square Root

Simplifying the square root, we get:

x=2±46x = \frac{2 \pm \sqrt{4}}{6}

Step 4: Simplify the Expression

Simplifying the expression, we get:

x=2±26x = \frac{2 \pm 2}{6}

Step 5: Solve for x

Solving for x, we get two possible solutions:

x=2+26=46=23x = \frac{2 + 2}{6} = \frac{4}{6} = \frac{2}{3}

x=2−26=06=0x = \frac{2 - 2}{6} = \frac{0}{6} = 0

Conclusion

In conclusion, the correct substitution of values from the given quadratic equation into the quadratic formula is:

x=−(−2)±(−2)2−4(3)(0)2(3)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(0)}}{2(3)}

Simplifying the expression, we get two possible solutions:

x=23x = \frac{2}{3}

x=0x = 0

Discussion

The quadratic formula is a powerful tool for solving quadratic equations, and it is essential to understand how to apply it correctly. In this article, we explored the correct substitution of values from a given quadratic equation into the quadratic formula. We simplified the expression and solved for x to get two possible solutions.

Common Mistakes

When substituting values into the quadratic formula, it is essential to pay attention to the signs and the order of operations. A common mistake is to substitute the values incorrectly, which can lead to incorrect solutions.

Tips and Tricks

When solving quadratic equations using the quadratic formula, it is essential to:

  • Pay attention to the signs and the order of operations.
  • Simplify the expression carefully.
  • Solve for x carefully.

By following these tips and tricks, you can ensure that you are solving quadratic equations correctly using the quadratic formula.

Real-World Applications

The quadratic formula has numerous 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 stress and strain on materials.
  • Economics: The quadratic formula is used to solve problems involving supply and demand.

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, and it has numerous real-world applications. By understanding how to apply the quadratic formula correctly, you can solve a wide range of problems in mathematics and other fields.

Final Thoughts

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 as follows:

x=−b±b2−4ac2ax = \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 substitute the values of a, b, and c into the formula. Then, simplify the expression and solve for x.

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

A: The values of a, b, and c are the coefficients of the quadratic equation. For example, in the quadratic equation 3x^2 + 2x + 1 = 0, the values are:

  • a = 3
  • b = 2
  • c = 1

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

A: To simplify the expression in the quadratic formula, you need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Evaluate any exponential expressions.
  3. Evaluate any multiplication and division expressions from left to right.
  4. Evaluate any addition and subtraction expressions from left to right.

Q: What is the difference between the two solutions in the quadratic formula?

A: The two solutions in the quadratic formula are the two possible values of x that satisfy the quadratic equation. The difference between the two solutions is the value of the expression under the square root, which is b^2 - 4ac.

Q: Can the quadratic formula be used to solve all types of quadratic equations?

A: No, the quadratic formula can only be used to solve quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are real numbers.

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:

  • Substituting the values of a, b, and c incorrectly.
  • Simplifying the expression incorrectly.
  • Failing to follow the order of operations.

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

A: The quadratic formula has numerous 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 stress and strain on materials.
  • Economics: The quadratic formula is used to solve problems involving supply and demand.

Q: How do I choose between the two solutions in the quadratic formula?

A: When choosing between the two solutions in the quadratic formula, you need to consider the context of the problem. For example, if you are solving a problem involving motion, you may need to choose the solution that corresponds to the initial velocity of the object.

Q: Can the quadratic formula be used to solve quadratic equations with complex coefficients?

A: No, the quadratic formula can only be used to solve quadratic equations with real coefficients. If the coefficients are complex, you need to use a different method to solve the equation.

Q: What are some tips for using the quadratic formula effectively?

A: Some tips for using the quadratic formula effectively include:

  • Paying attention to the signs and the order of operations.
  • Simplifying the expression carefully.
  • Solving for x carefully.
  • Checking your work to ensure that you have the correct solution.

By following these tips and avoiding common mistakes, you can use the quadratic formula effectively to solve quadratic equations.