Solve The Equation: $2z^2 - Z - 1 = 0$. Fully Simplify All Answers, Including Non-real Solutions.$z =$

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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 focus on solving the quadratic equation $2z^2 - z - 1 = 0$, which is a classic example of a quadratic equation with real coefficients. We will use the quadratic formula to find the solutions, and then simplify the answers to their most basic form.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $az^2 + bz + c = 0$, the solutions are given by:

z=βˆ’bΒ±b2βˆ’4ac2az = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our case, the equation is $2z^2 - z - 1 = 0$, so we have:

a = 2$, $b = -1$, and $c = -1

Applying the Quadratic Formula

Now that we have the values of $a$, $b$, and $c$, we can plug them into the quadratic formula:

z=βˆ’(βˆ’1)Β±(βˆ’1)2βˆ’4(2)(βˆ’1)2(2)z = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-1)}}{2(2)}

Simplifying the expression under the square root, we get:

z=1Β±1+84z = \frac{1 \pm \sqrt{1 + 8}}{4}

z=1Β±94z = \frac{1 \pm \sqrt{9}}{4}

z=1Β±34z = \frac{1 \pm 3}{4}

Simplifying the Solutions

Now that we have the two possible solutions, we can simplify them to their most basic form:

z=1+34=44=1z = \frac{1 + 3}{4} = \frac{4}{4} = 1

z=1βˆ’34=βˆ’24=βˆ’12z = \frac{1 - 3}{4} = \frac{-2}{4} = -\frac{1}{2}

Non-Real Solutions

In the previous section, we found two real solutions to the equation. However, in some cases, quadratic equations can have non-real solutions. These solutions are complex numbers, which have both real and imaginary parts.

To find the non-real solutions, we can use the quadratic formula again, but this time, we will use the fact that the square root of a negative number can be expressed as a complex number:

βˆ’1=i\sqrt{-1} = i

where $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$.

Complex Solutions

Using the quadratic formula, we can find the complex solutions to the equation:

z=βˆ’(βˆ’1)Β±(βˆ’1)2βˆ’4(2)(βˆ’1)2(2)z = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-1)}}{2(2)}

z=1Β±1+84z = \frac{1 \pm \sqrt{1 + 8}}{4}

z=1Β±94z = \frac{1 \pm \sqrt{9}}{4}

z=1Β±3i4z = \frac{1 \pm 3i}{4}

Simplifying the Complex Solutions

Now that we have the complex solutions, we can simplify them to their most basic form:

z=1+3i4z = \frac{1 + 3i}{4}

z=14+34iz = \frac{1}{4} + \frac{3}{4}i

z=14+34iz = \frac{1}{4} + \frac{3}{4}i

Conclusion

In this article, we solved the quadratic equation $2z^2 - z - 1 = 0$ using the quadratic formula. We found two real solutions, $z = 1$ and $z = -\frac{1}{2}$, and two complex solutions, $z = \frac{1}{4} + \frac{3}{4}i$ and $z = \frac{1}{4} - \frac{3}{4}i$. We simplified the solutions to their most basic form, and we hope that this article has provided a clear and concise guide to solving quadratic equations.

Final Answer

The final answer to the equation $2z^2 - z - 1 = 0$ is:

z = 1, -\frac{1}{2}, \frac{1}{4} + \frac{3}{4}i, \frac{1}{4} - \frac{3}{4}i$<br/> # **Quadratic Equation Q&A: Frequently Asked Questions and Answers**

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 solved the quadratic equation $2z^2 - z - 1 = 0$ using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations and provide additional information to help you better understand this topic.

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 (in this case, $z$) is two. The general form of a quadratic equation is $az^2 + bz + c = 0$, 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 using the quadratic formula. The quadratic formula is a powerful tool that can be used to solve any quadratic equation, and it is the method we used to solve the equation $2z^2 - z - 1 = 0$.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is given by:

z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} </span></p> <p>where $a$, $b$, and $c$ are the constants in the quadratic equation.</p> <h2><strong>Q: What are the steps to use the quadratic formula?</strong></h2> <p>A: To use the quadratic formula, you need to follow these steps:</p> <ol> <li>Identify the values of $a$, $b$, and $c$ in the quadratic equation.</li> <li>Plug these values into the quadratic formula.</li> <li>Simplify the expression under the square root.</li> <li>Simplify the solutions to their most basic form.</li> </ol> <h2><strong>Q: What are the different types of solutions to a quadratic equation?</strong></h2> <p>A: There are two types of solutions to a quadratic equation: real solutions and complex solutions. Real solutions are solutions that can be expressed as a single number, while complex solutions are solutions that have both real and imaginary parts.</p> <h2><strong>Q: How do I determine if a quadratic equation has real or complex solutions?</strong></h2> <p>A: To determine if a quadratic equation has real or complex solutions, you need to examine the expression under the square root in the quadratic formula. If the expression is positive, then the equation has real solutions. If the expression is negative, then the equation has complex solutions.</p> <h2><strong>Q: What are some common mistakes to avoid when solving quadratic equations?</strong></h2> <p>A: Some common mistakes to avoid when solving quadratic equations include:</p> <ul> <li>Not simplifying the expression under the square root.</li> <li>Not simplifying the solutions to their most basic form.</li> <li>Not checking if the solutions are real or complex.</li> <li>Not using the correct values of $a$, $b$, and $c$ in the quadratic formula.</li> </ul> <h2><strong>Q: How can I practice solving quadratic equations?</strong></h2> <p>A: There are several ways to practice solving quadratic equations, including:</p> <ul> <li>Using online resources, such as quadratic equation solvers and practice problems.</li> <li>Working with a tutor or teacher to practice solving quadratic equations.</li> <li>Using a calculator or computer program to solve quadratic equations.</li> <li>Creating your own practice problems and solving them.</li> </ul> <h2><strong>Conclusion</strong></h2> <p>In this article, we answered some frequently asked questions about quadratic equations and provided additional information to help you better understand this topic. We hope that this article has been helpful in clarifying any questions you may have had about quadratic equations.</p> <h2><strong>Final Answer</strong></h2> <p>The final answer to the equation $2z^2 - z - 1 = 0$ is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>z</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>βˆ’</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo separator="true">,</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>i</mi><mo separator="true">,</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>βˆ’</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>i</mi></mrow><annotation encoding="application/x-tex">z = 1, -\frac{1}{2}, \frac{1}{4} + \frac{3}{4}i, \frac{1}{4} - \frac{3}{4}i </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" 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