What Is The Solution?A. \[$x = -2500\$\]B. \[$x = -50\$\]C. \[$x = -2.5\$\]D. No Solution

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Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It has the general form of ax2+bx+c=0{ax^2 + bx + c = 0}, where a{a}, b{b}, and c{c} are constants, and x{x} is the variable. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}

This formula provides two solutions for the quadratic equation, which are the values of x{x} that satisfy the equation.

Example: Solving a Quadratic Equation

Let's consider the quadratic equation x2+5x+6=0{x^2 + 5x + 6 = 0}. To solve this equation, we can use the quadratic formula.

x=−5±52−4(1)(6)2(1){x = \frac{-5 \pm \sqrt{5^2 - 4(1)(6)}}{2(1)}}

x=−5±25−242{x = \frac{-5 \pm \sqrt{25 - 24}}{2}}

x=−5±12{x = \frac{-5 \pm \sqrt{1}}{2}}

x=−5±12{x = \frac{-5 \pm 1}{2}}

This gives us two possible solutions:

x=−5+12=−2{x = \frac{-5 + 1}{2} = -2}

x=−5−12=−3{x = \frac{-5 - 1}{2} = -3}

Therefore, the solutions to the quadratic equation x2+5x+6=0{x^2 + 5x + 6 = 0} are x=−2{x = -2} and x=−3{x = -3}.

What is the Solution to a Quadratic Equation?

The solution to a quadratic equation is the value or values of x{x} that satisfy the equation. In other words, it is the value or values of x{x} that make the equation true.

Types of Solutions

There are three types of solutions to a quadratic equation:

  1. Real and distinct solutions: These are solutions that are real numbers and are distinct from each other. In other words, they are not equal to each other.
  2. Real and equal solutions: These are solutions that are real numbers and are equal to each other. In other words, they are the same value.
  3. Complex solutions: These are solutions that are complex numbers. In other words, they have an imaginary part.

Real and Distinct Solutions

Real and distinct solutions are the most common type of solution to a quadratic equation. They are solutions that are real numbers and are distinct from each other.

Example: Real and Distinct Solutions

Let's consider the quadratic equation x2+4x+4=0{x^2 + 4x + 4 = 0}. To solve this equation, we can use the quadratic formula.

x=−4±42−4(1)(4)2(1){x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)}}

x=−4±16−162{x = \frac{-4 \pm \sqrt{16 - 16}}{2}}

x=−4±02{x = \frac{-4 \pm \sqrt{0}}{2}}

x=−42=−2{x = \frac{-4}{2} = -2}

Therefore, the solution to the quadratic equation x2+4x+4=0{x^2 + 4x + 4 = 0} is x=−2{x = -2}.

Real and Equal Solutions

Real and equal solutions are solutions that are real numbers and are equal to each other. In other words, they are the same value.

Example: Real and Equal Solutions

Let's consider the quadratic equation x2+2x+1=0{x^2 + 2x + 1 = 0}. To solve this equation, we can use the quadratic formula.

x=−2±22−4(1)(1)2(1){x = \frac{-2 \pm \sqrt{2^2 - 4(1)(1)}}{2(1)}}

x=−2±4−42{x = \frac{-2 \pm \sqrt{4 - 4}}{2}}

x=−2±02{x = \frac{-2 \pm \sqrt{0}}{2}}

x=−22=−1{x = \frac{-2}{2} = -1}

Therefore, the solution to the quadratic equation x2+2x+1=0{x^2 + 2x + 1 = 0} is x=−1{x = -1}.

Complex Solutions

Complex solutions are solutions that are complex numbers. In other words, they have an imaginary part.

Example: Complex Solutions

Let's consider the quadratic equation x2+1=0{x^2 + 1 = 0}. To solve this equation, we can use the quadratic formula.

x=−1±12−4(1)(1)2(1){x = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2(1)}}

x=−1±1−42{x = \frac{-1 \pm \sqrt{1 - 4}}{2}}

x=−1±−32{x = \frac{-1 \pm \sqrt{-3}}{2}}

x=−1±i32{x = \frac{-1 \pm i\sqrt{3}}{2}}

Therefore, the solutions to the quadratic equation x2+1=0{x^2 + 1 = 0} are x=−1+i32{x = \frac{-1 + i\sqrt{3}}{2}} and x=−1−i32{x = \frac{-1 - i\sqrt{3}}{2}}.

Conclusion

In conclusion, the solution to a quadratic equation is the value or values of x{x} that satisfy the equation. There are three types of solutions: real and distinct solutions, real and equal solutions, and complex solutions. Real and distinct solutions are the most common type of solution, while real and equal solutions are solutions that are real numbers and are equal to each other. Complex solutions are solutions that are complex numbers and have an imaginary part.

What is the Solution to a Quadratic Equation?

The solution to a quadratic equation is the value or values of x{x} that satisfy the equation. In other words, it is the value or values of x{x} that make the equation true.

Types of Solutions

There are three types of solutions to a quadratic equation:

  1. Real and distinct solutions: These are solutions that are real numbers and are distinct from each other. In other words, they are not equal to each other.
  2. Real and equal solutions: These are solutions that are real numbers and are equal to each other. In other words, they are the same value.
  3. Complex solutions: These are solutions that are complex numbers. In other words, they have an imaginary part.

Real and Distinct Solutions

Real and distinct solutions are the most common type of solution to a quadratic equation. They are solutions that are real numbers and are distinct from each other.

Example: Real and Distinct Solutions

Let's consider the quadratic equation x2+4x+4=0{x^2 + 4x + 4 = 0}. To solve this equation, we can use the quadratic formula.

x=−4±42−4(1)(4)2(1){x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)}}

x=−4±16−162{x = \frac{-4 \pm \sqrt{16 - 16}}{2}}

x=−4±02{x = \frac{-4 \pm \sqrt{0}}{2}}

x=−42=−2{x = \frac{-4}{2} = -2}

Therefore, the solution to the quadratic equation x2+4x+4=0{x^2 + 4x + 4 = 0} is x=−2{x = -2}.

Real and Equal Solutions

Real and equal solutions are solutions that are real numbers and are equal to each other. In other words, they are the same value.

Example: Real and Equal Solutions

Let's consider the quadratic equation x2+2x+1=0{x^2 + 2x + 1 = 0}. To solve this equation, we can use the quadratic formula.

x=−2±22−4(1)(1)2(1){x = \frac{-2 \pm \sqrt{2^2 - 4(1)(1)}}{2(1)}}

x=−2±4−42{x = \frac{-2 \pm \sqrt{4 - 4}}{2}}

x=−2±02{x = \frac{-2 \pm \sqrt{0}}{2}}

x=−22=−1{x = \frac{-2}{2} = -1}

Therefore, the solution to the quadratic equation x2+2x+1=0{x^2 + 2x + 1 = 0} is x=−1{x = -1}.

Complex Solutions

Complex solutions are solutions that are complex numbers. In other words, they have an imaginary part.

Example: Complex Solutions

Let's consider the quadratic equation x2+1=0{x^2 + 1 = 0}. To solve this equation, we can use the quadratic formula.

x=−1±12−4(1)(1)2(1){x = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2(1)}}

x=−1±1−42{x = \frac{-1 \pm \sqrt{1 - 4}}{2}}

x=−1±−32{x = \frac{-1 \pm \sqrt{-3}}{2}}

x=−1±i32{x = \frac{-1 \pm i\sqrt{3}}{2}}

Therefore, the solutions to the quadratic equation x2+1=0{x^2 + 1 = 0} are x=−1+i32{x = \frac{-1 + i\sqrt{3}}{2}} and x=−1−i32{x = \frac{-1 - i\sqrt{3}}{2}}.

Conclusion

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 is two. It has the general form of ax2+bx+c=0{ax^2 + bx + c = 0}, where a{a}, b{b}, and c{c} are constants, and x{x} is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}

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

A: There are three types of solutions to a quadratic equation:

  1. Real and distinct solutions: These are solutions that are real numbers and are distinct from each other. In other words, they are not equal to each other.
  2. Real and equal solutions: These are solutions that are real numbers and are equal to each other. In other words, they are the same value.
  3. Complex solutions: These are solutions that are complex numbers. In other words, they have an imaginary part.

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 can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real and distinct solutions. If the discriminant is zero, the equation has one real and equal solution. If the discriminant is negative, the equation has two complex solutions.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by:

b2−4ac{b^2 - 4ac}

Q: How do I use the discriminant to determine the type of solution?

A: To use the discriminant to determine the type of solution, you can follow these steps:

  1. Calculate the discriminant by substituting the values of a{a}, b{b}, and c{c} into the expression b2−4ac{b^2 - 4ac}.
  2. If the discriminant is positive, the equation has two real and distinct solutions.
  3. If the discriminant is zero, the equation has one real and equal solution.
  4. If the discriminant is negative, the equation has two complex solutions.

Q: What are some examples of quadratic equations with different types of solutions?

A: Here are some examples of quadratic equations with different types of solutions:

  • Real and distinct solutions: x2+4x+4=0{x^2 + 4x + 4 = 0}
  • Real and equal solutions: x2+2x+1=0{x^2 + 2x + 1 = 0}
  • Complex solutions: x2+1=0{x^2 + 1 = 0}

Q: How do I solve a quadratic equation with complex solutions?

A: To solve a quadratic equation with complex solutions, you can use the quadratic formula and simplify the expression to obtain the complex solutions.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Here are some common mistakes to avoid when solving quadratic equations:

  • Not using the correct method: Make sure to use the correct method for solving the quadratic equation, such as factoring, completing the square, or the quadratic formula.
  • Not simplifying the expression: Make sure to simplify the expression to obtain the correct solutions.
  • Not checking the solutions: Make sure to check the solutions to ensure that they are correct.

Conclusion

In conclusion, solving quadratic equations can be a challenging task, but with the right tools and techniques, it can be done with ease. By understanding the different types of solutions and how to determine them, you can solve quadratic equations with confidence. Remember to use the correct method, simplify the expression, and check the solutions to ensure that you obtain the correct solutions.