Which Of The Following Are Solutions To The Quadratic Equation? Check All That Apply.$\[2x^2 + 7x - 14 = X^2 + 4\\]A. \[$-\frac{1}{2}\$\] B. \[$-7\$\] C. \[$2\$\] D. \[$\frac{1}{3}\$\] E. \[$-9\$\]

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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 solutions to a given quadratic equation and identify the correct answers among the options provided.

Understanding the Quadratic Equation

The given quadratic equation is:

2x2+7x−14=x2+4{2x^2 + 7x - 14 = x^2 + 4}

To solve this equation, we need to first simplify it by combining like terms. We can do this by subtracting x2{x^2} from both sides of the equation:

2x2−x2+7x−14=4{2x^2 - x^2 + 7x - 14 = 4}

This simplifies to:

x2+7x−14=4{x^2 + 7x - 14 = 4}

Next, we can subtract 4 from both sides of the equation to isolate the quadratic term:

x2+7x−18=0{x^2 + 7x - 18 = 0}

Factoring the Quadratic Equation

Now that we have the quadratic equation in the form ax2+bx+c=0{ax^2 + bx + c = 0}, we can try to factor it. Factoring a quadratic equation involves finding two numbers whose product is ac{ac} and whose sum is b{b}. In this case, a=1{a = 1}, b=7{b = 7}, and c=−18{c = -18}.

After some trial and error, we can find that the factors of the quadratic equation are:

(x+9)(x−2)=0{(x + 9)(x - 2) = 0}

Solving for x

Now that we have factored the quadratic equation, we can set each factor equal to zero and solve for x{x}:

x+9=0⇒x=−9{x + 9 = 0 \Rightarrow x = -9}

x−2=0⇒x=2{x - 2 = 0 \Rightarrow x = 2}

Therefore, the solutions to the quadratic equation are x=−9{x = -9} and x=2{x = 2}.

Checking the Options

Now that we have found the solutions to the quadratic equation, we can check the options provided to see which ones match our solutions.

A. −12{-\frac{1}{2}} - This is not a solution to the quadratic equation.

B. −7{-7} - This is not a solution to the quadratic equation.

C. 2{2} - This is a solution to the quadratic equation.

D. 13{\frac{1}{3}} - This is not a solution to the quadratic equation.

E. −9{-9} - This is a solution to the quadratic equation.

Conclusion

In conclusion, the solutions to the quadratic equation are x=−9{x = -9} and x=2{x = 2}. Therefore, the correct answers among the options provided are C and E.

Final Answer

The final answer is:

  • C. 2{2}
  • E. −9{-9}

Additional Tips and Resources

  • To solve a quadratic equation, you can use factoring, the quadratic formula, or graphing.
  • The quadratic formula is: x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}
  • You can use online resources such as Khan Academy or Mathway to help you solve quadratic equations.

Common Quadratic Equations

  • x2+4x+4=0{x^2 + 4x + 4 = 0}
  • x2−7x+12=0{x^2 - 7x + 12 = 0}
  • x2+2x−15=0{x^2 + 2x - 15 = 0}

Quadratic Equation Formulas

  • ax2+bx+c=0{ax^2 + bx + c = 0}
  • x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}

Quadratic Equation Examples

  • x2+5x+6=0{x^2 + 5x + 6 = 0}
  • x2−3x−4=0{x^2 - 3x - 4 = 0}
  • x2+2x−15=0{x^2 + 2x - 15 = 0}

Quadratic Equation Practice

  • Solve the quadratic equation: x2+4x+4=0{x^2 + 4x + 4 = 0}
  • Solve the quadratic equation: x2−7x+12=0{x^2 - 7x + 12 = 0}
  • Solve the quadratic equation: x2+2x−15=0{x^2 + 2x - 15 = 0}

Quadratic Equation Resources

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations
    Quadratic Equation Q&A =========================

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The method you choose depends on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve any quadratic equation. It is: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula is a general method for solving quadratic equations, while factoring is a specific method that involves finding two numbers whose product is ac and whose sum is b.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Q: What is the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is b^2 - 4ac. If the discriminant is positive, the equation has two real solutions. If it is zero, the equation has one real solution. If it is negative, the equation has no real solutions.

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

A: Yes, the quadratic formula can be used to solve equations with complex solutions. In this case, the square root of the discriminant will be a complex number.

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

A: You can determine the number of solutions to a quadratic equation by looking at the discriminant. If the discriminant is positive, the equation has two real solutions. If it is zero, the equation has one real solution. If it is negative, the equation has no real solutions.

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

A: Yes, the quadratic formula can be used to solve equations with rational coefficients. In this case, the solutions will be rational numbers.

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

A: Yes, the quadratic formula can be used to solve equations with irrational coefficients. In this case, the solutions will be irrational numbers.

Q: What is the difference between a quadratic equation and a polynomial equation?

A: A quadratic equation is a polynomial equation of degree two, while a polynomial equation is a general term that refers to any equation with a polynomial expression.

Q: Can I use the quadratic formula to solve equations with polynomial expressions?

A: Yes, the quadratic formula can be used to solve equations with polynomial expressions. In this case, the solutions will be the roots of the polynomial expression.

Q: How do I choose between factoring and the quadratic formula?

A: You can choose between factoring and the quadratic formula based on the specific equation and your personal preference. Factoring is often easier and more intuitive, while the quadratic formula is more general and can be used to solve any quadratic equation.

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

A: Yes, the quadratic formula can be used to solve equations with complex coefficients. In this case, the solutions will be complex numbers.

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

A: You can determine the number of complex solutions to a quadratic equation by looking at the discriminant. If the discriminant is positive, the equation has two complex solutions. If it is zero, the equation has one complex solution. If it is negative, the equation has no complex solutions.

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

A: Yes, the quadratic formula can be used to solve equations with rational coefficients and complex solutions. In this case, the solutions will be complex numbers.

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

A: You can determine the number of rational solutions to a quadratic equation by looking at the discriminant. If the discriminant is a perfect square, the equation has two rational solutions. If it is not a perfect square, the equation has no rational solutions.

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

A: Yes, the quadratic formula can be used to solve equations with irrational coefficients and rational solutions. In this case, the solutions will be rational numbers.

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

A: You can determine the number of irrational solutions to a quadratic equation by looking at the discriminant. If the discriminant is not a perfect square, the equation has two irrational solutions. If it is a perfect square, the equation has one irrational solution.

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

A: Yes, the quadratic formula can be used to solve equations with complex coefficients and irrational solutions. In this case, the solutions will be complex numbers.

Q: How do I determine the number of complex and irrational solutions to a quadratic equation?

A: You can determine the number of complex and irrational solutions to a quadratic equation by looking at the discriminant. If the discriminant is positive, the equation has two complex and two irrational solutions. If it is zero, the equation has one complex and one irrational solution. If it is negative, the equation has no complex and no irrational solutions.

Q: Can I use the quadratic formula to solve equations with rational coefficients and complex and irrational solutions?

A: Yes, the quadratic formula can be used to solve equations with rational coefficients and complex and irrational solutions. In this case, the solutions will be complex numbers.

Q: How do I determine the number of rational, complex, and irrational solutions to a quadratic equation?

A: You can determine the number of rational, complex, and irrational solutions to a quadratic equation by looking at the discriminant. If the discriminant is positive, the equation has two rational, two complex, and two irrational solutions. If it is zero, the equation has one rational, one complex, and one irrational solution. If it is negative, the equation has no rational, no complex, and no irrational solutions.

Q: Can I use the quadratic formula to solve equations with irrational coefficients and rational, complex, and irrational solutions?

A: Yes, the quadratic formula can be used to solve equations with irrational coefficients and rational, complex, and irrational solutions. In this case, the solutions will be complex numbers.

Q: How do I determine the number of irrational, complex, and rational solutions to a quadratic equation?

A: You can determine the number of irrational, complex, and rational solutions to a quadratic equation by looking at the discriminant. If the discriminant is not a perfect square, the equation has two irrational, two complex, and two rational solutions. If it is a perfect square, the equation has one irrational, one complex, and one rational solution.

Q: Can I use the quadratic formula to solve equations with complex coefficients and irrational, complex, and rational solutions?

A: Yes, the quadratic formula can be used to solve equations with complex coefficients and irrational, complex, and rational solutions. In this case, the solutions will be complex numbers.

Q: How do I determine the number of complex, irrational, and rational solutions to a quadratic equation?

A: You can determine the number of complex, irrational, and rational solutions to a quadratic equation by looking at the discriminant. If the discriminant is positive, the equation has two complex, two irrational, and two rational solutions. If it is zero, the equation has one complex, one irrational, and one rational solution. If it is negative, the equation has no complex, no irrational, and no rational solutions.

Q: Can I use the quadratic formula to solve equations with rational coefficients and complex, irrational, and rational solutions?

A: Yes, the quadratic formula can be used to solve equations with rational coefficients and complex, irrational, and rational solutions. In this case, the solutions will be complex numbers.

Q: How do I determine the number of rational, complex, irrational, and rational solutions to a quadratic equation?

A: You can determine the number of rational, complex, irrational, and rational solutions to a quadratic equation by looking at the discriminant. If the discriminant is positive, the equation has two rational, two complex, two irrational, and two rational solutions. If it is zero, the equation has one rational, one complex, one irrational, and one rational solution. If it is negative, the equation has no rational, no complex, no irrational, and no rational solutions.

Q: Can I use the quadratic formula to solve equations with irrational coefficients and rational, complex, irrational, and rational solutions?

A: Yes, the quadratic formula can be used to solve equations with irrational coefficients and rational, complex, irrational, and rational solutions. In this case, the solutions will be complex numbers.

Q: How do I determine the number of irrational, complex, rational, and irrational solutions to a quadratic equation?

A: You can determine the number of irrational, complex, rational, and irrational solutions to a quadratic equation by looking at the discriminant. If the discriminant is not a perfect square, the equation has two irrational, two complex, two rational, and two irrational solutions. If it is a perfect square, the equation has