Find The Solutions To The Equation Below. Check All That Apply.$\[ 4x^2 + 4x + 1 = 0 \\]A. $\[ X = \frac{1}{2} \\]B. $\[ X = -\frac{1}{2} \\]C. $\[ X = \frac{3}{2} \\]D. $\[ X = 3 \\]E. $\[ X = 2

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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 provide a step-by-step guide on how to solve it.

The Quadratic Equation

The given quadratic equation is:

4x2+4x+1=0{ 4x^2 + 4x + 1 = 0 }

This equation is in the standard form of a quadratic equation, which is:

ax2+bx+c=0{ ax^2 + bx + c = 0 }

where a, b, and c are constants.

The Solutions

To solve the quadratic equation, we can use various methods, including factoring, completing the square, and the quadratic formula. In this case, we will use the quadratic formula.

The quadratic formula is given by:

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

where a, b, and c are the constants in the quadratic equation.

Step 1: Identify the Constants

In the given quadratic equation, we have:

a = 4 b = 4 c = 1

Step 2: Plug in the Values

Now, we can plug in the values of a, b, and c into the quadratic formula:

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

Step 3: Simplify the Expression

Simplifying the expression, we get:

x=−4±16−168{ x = \frac{-4 \pm \sqrt{16 - 16}}{8} }

x=−4±08{ x = \frac{-4 \pm \sqrt{0}}{8} }

x=−48{ x = \frac{-4}{8} }

x=−12{ x = -\frac{1}{2} }

The Final Answer

Therefore, the solution to the quadratic equation is:

x=−12{ x = -\frac{1}{2} }

Conclusion

In this article, we have solved a quadratic equation using the quadratic formula. We have identified the constants, plugged in the values, and simplified the expression to find the solution. The solution to the quadratic equation is x = -\frac{1}{2}.

Discussion

Now, let's discuss the solutions to the quadratic equation.

A. ${ x = \frac{1}{2} }$

This is not a solution to the quadratic equation.

B. ${ x = -\frac{1}{2} }$

This is a solution to the quadratic equation.

C. ${ x = \frac{3}{2} }$

This is not a solution to the quadratic equation.

D. ${ x = 3 }$

This is not a solution to the quadratic equation.

E. ${ x = 2 }$

This is not a solution to the quadratic equation.

Therefore, the correct answer is:

B. ${ x = -\frac{1}{2} }$

Additional Tips and Tricks

Here are some additional tips and tricks for solving quadratic equations:

  • Make sure to identify the constants correctly.
  • Plug in the values correctly into the quadratic formula.
  • Simplify the expression carefully.
  • Check your solution by plugging it back into the original equation.

By following these tips and tricks, you can solve quadratic equations with ease.

Conclusion

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we explored the solutions to a given quadratic equation and provided a step-by-step guide on how to solve it. In this article, we will answer some frequently asked questions about quadratic equations.

Q&A

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, completing the square, and the quadratic formula. The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the constants in the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the constants in the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the constants a, b, and c in the quadratic equation. Then, plug in the values of a, b, and c into the quadratic formula and simplify the expression.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring if it can be factored into the product of two binomials. For example, x^2 + 5x + 6 = (x + 3)(x + 2) = 0.

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 quadratic equation has two distinct real solutions. If the discriminant is zero, the quadratic equation has one real solution. If the discriminant is negative, the quadratic equation has no real solutions.

Q: Can I solve a quadratic equation by completing the square?

A: Yes, you can solve a quadratic equation by completing the square if it can be rewritten in the form (x + p)^2 = q. For example, x^2 + 6x + 8 = (x + 3)^2 - 1 = 0.

Q: What is the significance of the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It can be used to find the solutions to quadratic equations that cannot be factored or completed the square.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By using the quadratic formula and following the steps outlined in this article, you can solve quadratic equations with ease. Remember to identify the constants correctly, plug in the values correctly, simplify the expression carefully, and check your solution by plugging it back into the original equation.

Additional Tips and Tricks

Here are some additional tips and tricks for solving quadratic equations:

  • Make sure to identify the constants correctly.
  • Plug in the values correctly into the quadratic formula.
  • Simplify the expression carefully.
  • Check your solution by plugging it back into the original equation.
  • Use the quadratic formula to find the solutions to quadratic equations that cannot be factored or completed the square.
  • Use the discriminant to determine the number of real solutions to a quadratic equation.

By following these tips and tricks, you can solve quadratic equations with ease.