Which Of The Following Represents All Solutions Of $x^2 - 4x - 1 = 0$?1. $2 \pm \sqrt{5}$ 2. $-2 \pm \sqrt{5}$ 3. $2 \pm \sqrt{10}$ 4. $-2 \pm \sqrt{12}$

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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 process of solving quadratic equations, with a focus on the given equation $x^2 - 4x - 1 = 0$. We will examine the different methods of solving quadratic equations, including factoring, the quadratic formula, and completing the square. By the end of this article, you will have a comprehensive understanding of how to solve quadratic equations and be able to apply this knowledge to a variety of problems.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is a general formula that can be used to solve any quadratic equation of the form $ax^2 + bx + c = 0$. The quadratic formula is given by:

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

To use the quadratic formula, we need to identify the values of aa, bb, and cc in the given equation. In the equation $x^2 - 4x - 1 = 0$, we have a=1a = 1, b=−4b = -4, and c=−1c = -1. Plugging these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

x=4±16+42x = \frac{4 \pm \sqrt{16 + 4}}{2}

x=4±202x = \frac{4 \pm \sqrt{20}}{2}

x=4±252x = \frac{4 \pm 2\sqrt{5}}{2}

x=2±5x = 2 \pm \sqrt{5}

Factoring

Another method for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. In the equation $x^2 - 4x - 1 = 0$, we can try to factor the left-hand side as follows:

(x−a)(x−b)=x2−(a+b)x+ab(x - a)(x - b) = x^2 - (a + b)x + ab

We need to find two numbers aa and bb such that a+b=4a + b = 4 and ab=−1ab = -1. By trial and error, we can find that a=2a = 2 and b=−1b = -1 satisfy these conditions. Therefore, we can factor the left-hand side of the equation as follows:

(x−2)(x+1)=x2−4x−1(x - 2)(x + 1) = x^2 - 4x - 1

Setting each factor equal to zero, we get:

x−2=0orx+1=0x - 2 = 0 \quad \text{or} \quad x + 1 = 0

Solving for xx, we get:

x=2orx=−1x = 2 \quad \text{or} \quad x = -1

However, we notice that the solutions x=2x = 2 and x=−1x = -1 do not match the solutions obtained using the quadratic formula. This is because the factoring method only gives us the roots of the equation, but not the solutions in the form of x=a±bx = a \pm \sqrt{b}.

Completing the Square

Completing the square is another method for solving quadratic equations. This method involves rewriting the quadratic equation in the form x2+bx+c=(x+d)2+ex^2 + bx + c = (x + d)^2 + e. To complete the square, we need to add and subtract a constant term to the left-hand side of the equation. In the equation $x^2 - 4x - 1 = 0$, we can add and subtract (4/2)2=4(4/2)^2 = 4 to the left-hand side as follows:

x2−4x−1=(x2−4x+4)−4−1x^2 - 4x - 1 = (x^2 - 4x + 4) - 4 - 1

x2−4x−1=(x−2)2−5x^2 - 4x - 1 = (x - 2)^2 - 5

Setting the left-hand side equal to zero, we get:

(x−2)2−5=0(x - 2)^2 - 5 = 0

Adding 5 to both sides, we get:

(x−2)2=5(x - 2)^2 = 5

Taking the square root of both sides, we get:

x−2=±5x - 2 = \pm \sqrt{5}

Adding 2 to both sides, we get:

x=2±5x = 2 \pm \sqrt{5}

Conclusion

In this article, we have explored the process of solving quadratic equations, with a focus on the given equation $x^2 - 4x - 1 = 0$. We have examined the different methods of solving quadratic equations, including factoring, the quadratic formula, and completing the square. By using these methods, we have obtained the solutions x=2±5x = 2 \pm \sqrt{5}.

Which of the following represents all solutions of $x^2 - 4x - 1 = 0$?

Based on our analysis, we can conclude that the correct answer is:

  • 1. 2±52 \pm \sqrt{5}

This is because the solutions x=2±5x = 2 \pm \sqrt{5} are the only solutions that satisfy the equation $x^2 - 4x - 1 = 0$. The other options, −2±5-2 \pm \sqrt{5}, 2±102 \pm \sqrt{10}, and −2±12-2 \pm \sqrt{12}, do not satisfy the equation and are therefore incorrect.

Final Answer

The final answer is: 1\boxed{1}

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 process of solving quadratic equations, with a focus on the given equation $x^2 - 4x - 1 = 0$. We will examine the different methods of solving quadratic equations, including factoring, the quadratic formula, and completing the square. By the end of this article, you will have a comprehensive understanding of how to solve quadratic equations and be able to apply this knowledge to a variety of problems.

Q&A: Solving Quadratic Equations

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where aa, bb, and cc are constants.

Q: What are the different methods of solving quadratic equations?

A: There are several methods of solving quadratic equations, including:

  • Factoring: This involves expressing the quadratic equation as a product of two binomials.
  • The Quadratic Formula: This is a general formula that can be used to solve any quadratic equation of the form $ax^2 + bx + c = 0$.
  • Completing the Square: This involves rewriting the quadratic equation in the form x2+bx+c=(x+d)2+ex^2 + bx + c = (x + d)^2 + e.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to identify the values of aa, bb, and cc in the given equation. Then, you can plug these values into the quadratic formula:

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

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

A: The solutions obtained using the quadratic formula and factoring are the same, but they may be expressed in different forms. The quadratic formula gives you the solutions in the form of x=a±bx = a \pm \sqrt{b}, while factoring gives you the solutions in the form of x=ax = a or x=bx = b.

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

A: Yes, you can use completing the square to solve a quadratic equation. This involves rewriting the quadratic equation in the form x2+bx+c=(x+d)2+ex^2 + bx + c = (x + d)^2 + e. Then, you can set the left-hand side equal to zero and solve for xx.

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

A: Some common mistakes to avoid when solving quadratic equations include:

  • Not identifying the values of aa, bb, and cc correctly.
  • Not using the correct method for solving the equation.
  • Not checking the solutions for validity.

Conclusion

In this article, we have explored the process of solving quadratic equations, with a focus on the given equation $x^2 - 4x - 1 = 0$. We have examined the different methods of solving quadratic equations, including factoring, the quadratic formula, and completing the square. By using these methods, we have obtained the solutions x=2±5x = 2 \pm \sqrt{5}. We have also answered some common questions about solving quadratic equations, including how to use the quadratic formula and what are some common mistakes to avoid.

Final Answer

The final answer is: 1\boxed{1}