The Equation $4w^2 + 3w - 6 = 0$ Has Solutions Of The Form $w = \frac{N \pm \sqrt{D}}{M}$.(A) Solve This Equation And Find The Appropriate Values Of $N$, $M$, And $D$. Do Not Worry About Simplifying The

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Introduction

In this article, we will be solving the quadratic equation $4w^2 + 3w - 6 = 0$ and expressing its solutions in the form of $w = \frac{N \pm \sqrt{D}}{M}$. This form is a standard way of expressing the solutions to a quadratic equation, where $N$, $M$, and $D$ are constants that need to be determined.

The Quadratic Formula

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

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

In our case, the equation is $4w^2 + 3w - 6 = 0$, so we have $a = 4$, $b = 3$, and $c = -6$. Plugging these values into the quadratic formula, we get:

$$w = \frac{-3 \pm \sqrt{3^2 - 4(4)(-6)}}{2(4)}$$

Simplifying the Expression

Now, let's simplify the expression under the square root. We have:

$$3^2 - 4(4)(-6) = 9 + 96 = 105$$

So, the expression becomes:

$$w = \frac{-3 \pm \sqrt{105}}{8}$$

Determining the Values of $N$, $M$, and $D$

Comparing this expression with the standard form $w = \frac{N \pm \sqrt{D}}{M}$, we can see that:

• $N = -3$
• $M = 8$
• $D = 105$

Therefore, the values of $N$, $M$, and $D$ are $-3$, $8$, and $105$, respectively.

Conclusion

In this article, we have solved the quadratic equation $4w^2 + 3w - 6 = 0$ and expressed its solutions in the form of $w = \frac{N \pm \sqrt{D}}{M}$. We have determined the values of $N$, $M$, and $D$ to be $-3$, $8$, and $105$, respectively. This form is a standard way of expressing the solutions to a quadratic equation, and it provides a clear and concise way of presenting the solutions.

The Importance of the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It provides a general solution to any quadratic equation, and it can be used to solve equations with complex or irrational solutions. In addition, the quadratic formula can be used to solve equations with multiple solutions, such as repeated roots.

Real-World Applications of the Quadratic Formula

The quadratic formula has many real-world applications. For example, it can be used to model the trajectory of a projectile, such as a thrown ball or a rocket. It can also be used to solve problems in physics, engineering, and economics. In addition, the quadratic formula can be used to solve problems in computer science, such as finding the shortest path between two points in a graph.

Solving Quadratic Equations with Complex Solutions

In some cases, the quadratic formula may produce complex solutions. These solutions can be expressed in the form of $w = \frac{N \pm \sqrt{D}}{M}$, where $D$ is a negative number. In this case, the square root of $D$ is an imaginary number, and the solution is said to be complex.

Solving Quadratic Equations with Repeated Roots

In some cases, the quadratic formula may produce repeated roots. These roots can be expressed in the form of $w = \frac{N \pm \sqrt{D}}{M}$, where $D$ is zero. In this case, the square root of $D$ is zero, and the solution is said to be repeated.

Conclusion

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Introduction

In our previous article, we solved the quadratic equation $4w^2 + 3w - 6 = 0$ and expressed its solutions in the form of $w = \frac{N \pm \sqrt{D}}{M}$. In this article, we will answer some frequently asked questions about quadratic equation solutions.

Q: What is the quadratic formula?

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

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

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

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. For example, if you have the equation $4w^2 + 3w - 6 = 0$, you would plug in $a = 4$, $b = 3$, and $c = -6$ into the formula.

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

A: A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. A linear equation, on the other hand, is an equation of the form $ax + b = 0$, where $a$ and $b$ are constants.

Q: Can a quadratic equation have more than two solutions?

A: Yes, a quadratic equation can have more than two solutions. However, in most cases, a quadratic equation will have two distinct solutions.

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

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$. 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 repeated real solution. If the discriminant is negative, the equation has two complex solutions.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. This occurs when the discriminant is negative.

Q: How do I determine the nature of the solutions to a quadratic equation?

A: To determine the nature of the solutions to a quadratic equation, you need to calculate the discriminant. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one repeated real solution. If the discriminant is negative, the equation has two complex solutions.

Q: Can a quadratic equation have repeated roots?

A: Yes, a quadratic equation can have repeated roots. This occurs when the discriminant is zero.

Q: How do I solve a quadratic equation with repeated roots?

A: To solve a quadratic equation with repeated roots, you need to use the quadratic formula and set the discriminant to zero. This will give you a repeated root.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equation solutions. We have discussed the quadratic formula, the difference between quadratic and linear equations, and the significance of the discriminant. We have also discussed how to determine the nature of the solutions to a quadratic equation and how to solve a quadratic equation with repeated roots.