Identify The Values Of { A$}$, { B$}$, And { C$}$ That Could Be Used With The Quadratic Formula To Solve The Equation. Enter { A$}$ As A Positive Integer Value.Given Equation: [$x^2 = 5(x -

by ADMIN 190 views

Introduction

The quadratic formula is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0. In this article, we will explore how to identify the values of a, b, and c that could be used with the quadratic formula to solve the equation x^2 = 5(x - 2). We will focus on rewriting the equation in the standard form of a quadratic equation and then use the quadratic formula to find the solutions.

Rewriting the Equation

The given equation is x^2 = 5(x - 2). To rewrite this equation in the standard form of a quadratic equation, we need to expand the right-hand side and move all terms to one side of the equation.

x^2 = 5x - 10

Now, we can rewrite the equation as:

x^2 - 5x + 10 = 0

This is the standard form of a quadratic equation, where a = 1, b = -5, and c = 10.

Identifying Values for a, b, and c

Now that we have rewritten the equation in the standard form, we can identify the values of a, b, and c that could be used with the quadratic formula.

  • Value of a: The value of a is the coefficient of the x^2 term. In this case, a = 1.
  • Value of b: The value of b is the coefficient of the x term. In this case, b = -5.
  • Value of c: The value of c is the constant term. In this case, c = 10.

Using the Quadratic Formula

The quadratic formula is given by:

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

We can now use this formula to find the solutions to the equation x^2 - 5x + 10 = 0.

x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(10)}}{2(1)}

Simplifying the expression, we get:

x = \frac{5 \pm \sqrt{25 - 40}}{2}

Further simplifying, we get:

x = \frac{5 \pm \sqrt{-15}}{2}

Since the square root of a negative number is not a real number, this equation has no real solutions.

Conclusion

In this article, we have identified the values of a, b, and c that could be used with the quadratic formula to solve the equation x^2 = 5(x - 2). We have rewritten the equation in the standard form of a quadratic equation and then used the quadratic formula to find the solutions. We have found that the equation has no real solutions.

Tips and Variations

  • Using the quadratic formula with complex numbers: If the equation has complex solutions, we can use the quadratic formula to find the solutions.
  • Using the quadratic formula with rational coefficients: If the equation has rational solutions, we can use the quadratic formula to find the solutions.
  • Using the quadratic formula with integer coefficients: If the equation has integer solutions, we can use the quadratic formula to find the solutions.

Practice Problems

  • Solve the equation x^2 + 4x + 4 = 0 using the quadratic formula.
  • Solve the equation x^2 - 6x + 8 = 0 using the quadratic formula.
  • Solve the equation x^2 + 2x + 1 = 0 using the quadratic formula.

Glossary

  • Quadratic equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • Quadratic formula: A formula for solving quadratic equations of the form ax^2 + bx + c = 0.
  • Standard form: The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.

References

  • "Quadratic Equations" by Math Open Reference
  • "Quadratic Formula" by Khan Academy
  • "Quadratic Equations and Functions" by Purplemath
    Quadratic Equation Q&A =========================

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving 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 in the quadratic equation. Then, you can plug these values into the quadratic formula and simplify to find the solutions.

Q: What are the steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

  1. Identify the values of a, b, and c in the quadratic equation.
  2. Plug these values into the quadratic formula.
  3. Simplify the expression to find the solutions.

Q: What are the different types of solutions to a quadratic equation?

A: The different types of solutions to a quadratic equation are:

  • Real solutions: Solutions that are real numbers.
  • Complex solutions: Solutions that are complex numbers.
  • Rational solutions: Solutions that are rational numbers.
  • Integer solutions: Solutions that are integers.

Q: How do I determine the type of solution to a quadratic equation?

A: To determine the type of solution to a quadratic equation, you need to examine the discriminant (b^2 - 4ac). If the discriminant is:

  • Positive: The equation has two real solutions.
  • Zero: The equation has one real solution.
  • Negative: The equation has two complex solutions.

Q: What is the discriminant?

A: The discriminant is the expression b^2 - 4ac in the quadratic formula. It determines the type of solution to the quadratic equation.

Q: How do I simplify the quadratic formula?

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

  1. Evaluate the expressions inside the parentheses.
  2. Exponentiate the expressions (if any).
  3. Multiply and divide the expressions (if any).
  4. Add and subtract the expressions (if any).

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

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

  • Incorrectly identifying the values of a, b, and c.
  • Not following the order of operations.
  • Not simplifying the expression correctly.

Q: How do I check my work when using the quadratic formula?

A: To check your work when using the quadratic formula, you can:

  • Plug the solutions back into the original equation.
  • Verify that the solutions satisfy the equation.
  • Check that the solutions are correct.

Practice Problems

  • Solve the equation x^2 + 4x + 4 = 0 using the quadratic formula.
  • Solve the equation x^2 - 6x + 8 = 0 using the quadratic formula.
  • Solve the equation x^2 + 2x + 1 = 0 using the quadratic formula.

Glossary

  • Quadratic equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • Quadratic formula: A formula for solving quadratic equations of the form ax^2 + bx + c = 0.
  • Discriminant: The expression b^2 - 4ac in the quadratic formula.
  • Real solutions: Solutions that are real numbers.
  • Complex solutions: Solutions that are complex numbers.
  • Rational solutions: Solutions that are rational numbers.
  • Integer solutions: Solutions that are integers.

References

  • "Quadratic Equations" by Math Open Reference
  • "Quadratic Formula" by Khan Academy
  • "Quadratic Equations and Functions" by Purplemath