Consider The Quadratic Equation:$\[ -4z^2 - 12z + 4 = 0 \\]State The Values Of \[$a, B, C\$\] In The Standard Form Of A Quadratic Equation:- \[$a = -4\$\]- \[$b = -12\$\]- \[$c = 4\$\]Determine The Value Of The

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving a quadratic equation of the form $-4z^2 - 12z + 4 = 0$. We will start by identifying the values of $a$, $b$, and $c$ in the standard form of a quadratic equation, and then we will determine the value of the quadratic equation using the quadratic formula.

The Standard Form of a Quadratic Equation

The standard form of a quadratic equation is given by:

$$az^2 + bz + c = 0$$

where $a$, $b$, and $c$ are constants. In the given quadratic equation $-4z^2 - 12z + 4 = 0$, we can identify the values of $a$, $b$, and $c$ as follows:

• $a = -4$: This is the coefficient of the $z^2$ term.
• $b = -12$: This is the coefficient of the $z$ term.
• $c = 4$: This is the constant term.

The Quadratic Formula

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

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

where $a$, $b$, and $c$ are the constants in the quadratic equation. In our case, we have $a = -4$, $b = -12$, and $c = 4$. Plugging these values into the quadratic formula, we get:

$$z = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(-4)(4)}}{2(-4)}$$

Simplifying the Quadratic Formula

To simplify the quadratic formula, we can start by evaluating the expression inside the square root:

$$(-12)^2 - 4(-4)(4) = 144 + 64 = 208$$

Next, we can simplify the expression inside the square root:

$$\sqrt{208} = \sqrt{16 \cdot 13} = 4\sqrt{13}$$

Now, we can plug this value back into the quadratic formula:

$$z = \frac{-(-12) \pm 4\sqrt{13}}{2(-4)}$$

Simplifying the Quadratic Formula Further

To simplify the quadratic formula further, we can start by evaluating the expression inside the parentheses:

$$-(-12) = 12$$

Next, we can simplify the expression inside the square root:

$$4\sqrt{13} = 4\sqrt{13}$$

Now, we can plug this value back into the quadratic formula:

$$z = \frac{12 \pm 4\sqrt{13}}{-8}$$

Simplifying the Quadratic Formula Even Further

To simplify the quadratic formula even further, we can start by evaluating the expression inside the parentheses:

$$-8 = -8$$

Next, we can simplify the expression inside the square root:

$$4\sqrt{13} = 4\sqrt{13}$$

Now, we can plug this value back into the quadratic formula:

$$z = \frac{12 \pm 4\sqrt{13}}{-8}$$

The Final Answer

The final answer is:

$$z = \frac{12 \pm 4\sqrt{13}}{-8}$$

This is the solution to the quadratic equation $-4z^2 - 12z + 4 = 0$. We can see that the solution is a complex number, which is a common feature of quadratic equations.

Conclusion

In this article, we have solved a quadratic equation of the form $-4z^2 - 12z + 4 = 0$. We have identified the values of $a$, $b$, and $c$ in the standard form of a quadratic equation, and then we have determined the value of the quadratic equation using the quadratic formula. The solution to the quadratic equation is a complex number, which is a common feature of quadratic equations. We hope that this article has provided a clear and concise explanation of how to solve quadratic equations.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we solved a quadratic equation of the form $-4z^2 - 12z + 4 = 0$. In this article, we will answer some frequently asked questions about 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 $az^2 + bz + c = 0$, 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, the quadratic formula, and graphing. The quadratic formula is a powerful tool for solving quadratic equations, and it is given by:

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

Q: What is the quadratic formula?

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

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

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 inside the square root and solve for $z$.

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. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring. However, not all quadratic equations can be factored, and the quadratic formula is a more general method for solving quadratic equations.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression inside the square root in the quadratic formula, which is given by $b^2 - 4ac$. The discriminant determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: Can I use the quadratic formula to solve a cubic equation?

A: No, you cannot use the quadratic formula to solve a cubic equation. The quadratic formula is only applicable to quadratic equations, and it is not a general method for solving polynomial equations of higher degree.

Q: What is the relationship between the quadratic formula and the square root?

A: The quadratic formula involves the square root of the discriminant, which is given by $b^2 - 4ac$. The square root is a fundamental operation in mathematics, and it is used to find the square root of a number.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations. We hope that this article has provided a clear and concise explanation of the concepts and methods involved in solving quadratic equations. If you have any further questions, please feel free to ask.