Rhett Is Solving The Quadratic Equation 0 = X 2 − 2 X − 3 0 = X^2 - 2x - 3 0 = X 2 − 2 X − 3 Using The Quadratic Formula. Which Shows The Correct Substitution Of The Values A , B A, B A , B , And C C C Into The Quadratic Formula?Quadratic Formula: $x = \frac{-b \pm

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Understanding the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is given by the equation:

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

where aa, bb, and cc are the coefficients of the quadratic equation.

Identifying the Coefficients

In the given quadratic equation 0=x2−2x−30 = x^2 - 2x - 3, we can identify the coefficients as follows:

  • a=1a = 1 (coefficient of x2x^2)
  • b=−2b = -2 (coefficient of xx)
  • c=−3c = -3 (constant term)

Substituting the Values into the Quadratic Formula

To solve the quadratic equation using the quadratic formula, we need to substitute the values of aa, bb, and cc into the formula.

Option 1: Incorrect Substitution

One possible substitution is:

x=−(−2)±(−2)2−4(1)(−3)2(1)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)}

This substitution is incorrect because it does not follow the correct order of operations.

Option 2: Correct Substitution

The correct substitution is:

x=−(−2)±(−2)2−4(1)(−3)2(1)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)}

However, this can be simplified to:

x=2±4+122x = \frac{2 \pm \sqrt{4 + 12}}{2}

x=2±162x = \frac{2 \pm \sqrt{16}}{2}

x=2±42x = \frac{2 \pm 4}{2}

This gives us two possible solutions:

x=2+42=3x = \frac{2 + 4}{2} = 3

x=2−42=−1x = \frac{2 - 4}{2} = -1

Conclusion

In conclusion, the correct substitution of the values aa, bb, and cc into the quadratic formula is:

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

where a=1a = 1, b=−2b = -2, and c=−3c = -3.

The correct substitution is:

x=2±4+122x = \frac{2 \pm \sqrt{4 + 12}}{2}

x=2±162x = \frac{2 \pm \sqrt{16}}{2}

x=2±42x = \frac{2 \pm 4}{2}

This gives us two possible solutions:

x=2+42=3x = \frac{2 + 4}{2} = 3

x=2−42=−1x = \frac{2 - 4}{2} = -1

Therefore, the correct answer is:

The correct substitution of the values aa, bb, and cc into the quadratic formula is x=2±4+122x = \frac{2 \pm \sqrt{4 + 12}}{2}

Additional Tips and Tricks

  • When using the quadratic formula, make sure to follow the correct order of operations.
  • Simplify the expression under the square root before substituting the values.
  • Check your solutions by plugging them back into the original equation.

Real-World Applications

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

  • Physics: The quadratic formula is used to model the motion of objects under the influence of gravity.
  • 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 systems of linear equations.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By understanding the formula and how to substitute the values of aa, bb, and cc, we can solve quadratic equations and apply the solutions to real-world problems.

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Frequently Asked Questions

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is given by the equation:

x=−b±b2−4ac2ax = \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 coefficients aa, bb, and cc in the quadratic equation. Then, substitute these values into the formula and simplify the expression.

Q: What are the coefficients aa, bb, and cc?

A: The coefficients aa, bb, and cc are the numbers in front of the x2x^2, xx, and constant terms in the quadratic equation, respectively.

Q: How do I simplify the expression under the square root?

A: To simplify the expression under the square root, you need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Exponentiate (i.e., raise to a power).
  3. Multiply and divide from left to right.
  4. Add and subtract from left to right.

Q: What if the expression under the square root is negative?

A: If the expression under the square root is negative, then the quadratic equation has no real solutions. In this case, the quadratic formula will give you complex solutions.

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, the solutions will be complex numbers.

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original quadratic equation and simplify. If the equation is true, then the solution is correct.

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 model the motion of objects under the influence of gravity.
  • 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 systems of linear equations.

Q: Can I use the quadratic formula to solve systems of linear equations?

A: Yes, you can use the quadratic formula to solve systems of linear equations. However, you need to first convert the system into a quadratic equation.

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:

  • Not following the order of operations.
  • Not simplifying the expression under the square root.
  • Not checking the solutions.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By understanding the formula and how to substitute the values of aa, bb, and cc, we can solve quadratic equations and apply the solutions to real-world problems. Remember to follow the order of operations, simplify the expression under the square root, and check your solutions to avoid common mistakes.

Additional Resources

  • Quadratic Formula Calculator: A calculator that can help you solve quadratic equations using the quadratic formula.
  • Quadratic Formula Worksheet: A worksheet that provides practice problems for solving quadratic equations using the quadratic formula.
  • Quadratic Formula Video Tutorial: A video tutorial that explains how to use the quadratic formula to solve quadratic equations.