A) Show That The Solutions Of $ax^2 + Bx + C = 0$ Are $x = \frac{2a}{2a}$.b) Solve The Equation $x^{2/3} + X^{1/3} - 2 = 0$.c) (i) State Whether The Operation Is A Binary Operation On $ Z \mathbb{Z} Z [/tex],

Introduction

Quadratic equations and cubic equations are fundamental concepts in mathematics, and solving them is crucial in various fields such as physics, engineering, and computer science. In this article, we will show that the solutions of a quadratic equation $ax^2 + bx + c = 0$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, and then solve the equation $x^{2/3} + x^{1/3} - 2 = 0$. We will also discuss whether a given operation is a binary operation on the set of integers $\mathbb{Z}$.

Solving Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. To solve a quadratic equation, we can use the quadratic formula, which is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula is derived by using the method of completing the square, which involves manipulating the equation to express it in the form $(x + p)^2 = q$, where $p$ and $q$ are constants.

Derivation of the Quadratic Formula

To derive the quadratic formula, we start with the quadratic equation $ax^2 + bx + c = 0$. We can rewrite this equation as $ax^2 + bx = -c$. Then, we add $\left(\frac{b}{2a}\right)^2$ to both sides of the equation, which gives us:

$ax^2 + bx + \left(\frac{b}{2a}\right)^2 = -c + \left(\frac{b}{2a}\right)^2$

This can be rewritten as:

$a\left(x + \frac{b}{2a}\right)^2 = -c + \left(\frac{b}{2a}\right)^2$

Now, we can take the square root of both sides of the equation, which gives us:

$x + \frac{b}{2a} = \pm \sqrt{-c + \left(\frac{b}{2a}\right)^2}$

Subtracting $\frac{b}{2a}$ from both sides of the equation, we get:

$x = -\frac{b}{2a} \pm \sqrt{-c + \left(\frac{b}{2a}\right)^2}$

Simplifying the expression under the square root, we get:

$x = -\frac{b}{2a} \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}$

Finally, we can simplify the expression by multiplying both sides of the equation by $2a$, which gives us the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Solving the Equation $x^{2/3} + x^{1/3} - 2 = 0$

To solve the equation $x^{2/3} + x^{1/3} - 2 = 0$, we can use the method of substitution. Let $y = x^{1/3}$, which means $x = y^3$. Substituting this into the equation, we get:

$y^6 + y^3 - 2 = 0$

This is a cubic equation in $y$, and we can solve it using the method of factoring. Factoring the equation, we get:

$(y^3 + 2)(y^3 - 1) = 0$

This gives us two possible solutions: $y^3 + 2 = 0$ and $y^3 - 1 = 0$. Solving the first equation, we get:

$y^3 = -2$

$y = \sqrt[3]{-2}$

Solving the second equation, we get:

$y^3 = 1$

$y = 1$

Since $y = x^{1/3}$, we can substitute this back into the equation to get:

$x^{1/3} = \sqrt[3]{-2}$

$x^{1/3} = 1$

Solving for $x$, we get:

$x = -2$

$x = 1$

Therefore, the solutions to the equation $x^{2/3} + x^{1/3} - 2 = 0$ are $x = -2$ and $x = 1$.

Binary Operations on $\mathbb{Z}$

A binary operation on a set $S$ is a function that takes two elements from $S$ and produces another element from $S$. In other words, a binary operation is a way of combining two elements from a set to produce another element from the same set.

To determine whether a given operation is a binary operation on $\mathbb{Z}$, we need to check whether the operation satisfies the following properties:

1. Closure: The operation must produce an element from $\mathbb{Z}$ when applied to two elements from $\mathbb{Z}$.
2. Associativity: The operation must satisfy the associative property, which means that the order in which we apply the operation does not matter.
3. Existence of Identity: The operation must have an identity element, which means that there exists an element in $\mathbb{Z}$ that does not change the result when combined with any other element.

Let's consider the operation $*$ defined by $a * b = a + b$ for all $a, b \in \mathbb{Z}$. This operation satisfies the closure property, since the sum of two integers is always an integer. It also satisfies the associative property, since the order in which we add two integers does not matter. Finally, it has an identity element, which is $0$, since $a * 0 = a$ for all $a \in \mathbb{Z}$.

Therefore, the operation $*$ defined by $a * b = a + b$ is a binary operation on $\mathbb{Z}$.

Conclusion

In this article, we have shown that the solutions of a quadratic equation $ax^2 + bx + c = 0$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, and then solved the equation $x^{2/3} + x^{1/3} - 2 = 0$. We have also discussed whether a given operation is a binary operation on the set of integers $\mathbb{Z}$. We have shown that the operation $*$ defined by $a * b = a + b$ is a binary operation on $\mathbb{Z}$.

Introduction

Quadratic equations and cubic equations are fundamental concepts in mathematics, and solving them is crucial in various fields such as physics, engineering, and computer science. In this article, we will provide a Q&A guide to help you understand and solve quadratic equations and cubic equations.

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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \neq 0$.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula is derived by using the method of completing the square, which involves manipulating the equation to express it in the form $(x + p)^2 = q$, where $p$ and $q$ are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the solutions to a quadratic equation. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

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

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means the highest power of the variable is three. The general form of a cubic equation is $ax^3 + bx^2 + cx + d = 0$, where $a$, $b$, $c$, and $d$ are constants, and $a \neq 0$.

Q: How do I solve a cubic equation?

A: To solve a cubic equation, you can use various methods such as factoring, the rational root theorem, or the cubic formula. The cubic formula is a formula that gives the solutions to a cubic equation, but it is more complex than the quadratic formula.

Q: What is the cubic formula?

A: The cubic formula is a formula that gives the solutions to a cubic equation. It is given by:

$x = \frac{-b \pm \sqrt[3]{-q + \sqrt{q^2 + \frac{4p^3}{27}}}}{2}$

where $p = a$, $q = b$, and $r = c$.

Q: How do I use the cubic formula?

A: To use the cubic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, you can simplify the expression to get the solutions to the equation.

Q: What is a binary operation on $\mathbb{Z}$?

A: A binary operation on $\mathbb{Z}$ is a function that takes two elements from $\mathbb{Z}$ and produces another element from $\mathbb{Z}$. In other words, a binary operation is a way of combining two elements from a set to produce another element from the same set.

Q: How do I determine whether an operation is a binary operation on $\mathbb{Z}$?

A: To determine whether an operation is a binary operation on $\mathbb{Z}$, you need to check whether the operation satisfies the following properties:

1. Closure: The operation must produce an element from $\mathbb{Z}$ when applied to two elements from $\mathbb{Z}$.
2. Associativity: The operation must satisfy the associative property, which means that the order in which we apply the operation does not matter.
3. Existence of Identity: The operation must have an identity element, which means that there exists an element in $\mathbb{Z}$ that does not change the result when combined with any other element.

Conclusion

In this article, we have provided a Q&A guide to help you understand and solve quadratic equations and cubic equations. We have also discussed binary operations on $\mathbb{Z}$ and how to determine whether an operation is a binary operation on $\mathbb{Z}$. We hope that this guide has been helpful in understanding and solving quadratic equations and cubic equations.

Frequently Asked Questions

• Q: What is the difference between a quadratic equation and a cubic equation? A: A quadratic equation is a polynomial equation of degree two, while a cubic equation is a polynomial equation of degree three.
• Q: How do I solve a quadratic equation with complex roots? A: To solve a quadratic equation with complex roots, you can use the quadratic formula and then simplify the expression to get the complex roots.
• Q: What is the significance of the quadratic formula? A: The quadratic formula is a formula that gives the solutions to a quadratic equation, and it is used in various fields such as physics, engineering, and computer science.
• Q: How do I determine whether an operation is commutative on $\mathbb{Z}$? A: To determine whether an operation is commutative on $\mathbb{Z}$, you need to check whether the operation satisfies the commutative property, which means that the order in which we apply the operation does not matter.

Glossary

• Quadratic equation: A polynomial equation of degree two.
• Cubic equation: A polynomial equation of degree three.
• Binary operation: A function that takes two elements from a set and produces another element from the same set.
• Closure: The property that an operation must produce an element from a set when applied to two elements from the same set.
• Associativity: The property that an operation must satisfy the associative property, which means that the order in which we apply the operation does not matter.
• Existence of Identity: The property that an operation must have an identity element, which means that there exists an element in a set that does not change the result when combined with any other element.