Solve For The Roots In Simplest Form Using The Quadratic Formula: 4 X 2 + 29 = − 28 X 4x^2 + 29 = -28x 4 X 2 + 29 = − 28 X Solve For X X X : X = □ X = \square X = □

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 a quadratic equation using the quadratic formula, which is a powerful tool for finding the roots of a quadratic equation. We will use the equation $4x^2 + 29 = -28x$ as an example and solve for $x$ in its simplest form.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

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

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic formula is a general solution that works for all quadratic equations, regardless of whether they can be factored or not.

Solving the Quadratic Equation

Now, let's apply the quadratic formula to the equation $4x^2 + 29 = -28x$. First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting $29$ from both sides and adding $28x$ to both sides:

$4x^2 + 28x + 29 = 0$

Now, we can identify the coefficients $a$, $b$, and $c$:

$a = 4$, $b = 28$, and $c = 29$

Substituting the Coefficients into the Quadratic Formula

Now that we have identified the coefficients, we can substitute them into the quadratic formula:

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

Substituting $a = 4$, $b = 28$, and $c = 29$, we get:

$x = \frac{-28 \pm \sqrt{(28)^2 - 4(4)(29)}}{2(4)}$

Simplifying the Expression

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

$(28)^2 - 4(4)(29) = 784 - 464 = 320$

So, the expression becomes:

$x = \frac{-28 \pm \sqrt{320}}{8}$

Simplifying the Square Root

The square root of $320$ can be simplified as follows:

$\sqrt{320} = \sqrt{64 \times 5} = 8\sqrt{5}$

So, the expression becomes:

$x = \frac{-28 \pm 8\sqrt{5}}{8}$

Simplifying the Expression Further

Now, let's simplify the expression further by dividing both the numerator and the denominator by $8$:

$x = \frac{-28}{8} \pm \frac{8\sqrt{5}}{8}$

$x = -\frac{28}{8} \pm \frac{8\sqrt{5}}{8}$

$x = -\frac{7}{2} \pm \frac{\sqrt{5}}{1}$

The Final Answer

Therefore, the roots of the quadratic equation $4x^2 + 29 = -28x$ are:

$x = -\frac{7}{2} + \frac{\sqrt{5}}{1}$ and $x = -\frac{7}{2} - \frac{\sqrt{5}}{1}$

These are the simplest forms of the roots, and they can be further simplified if needed.

Conclusion

Introduction

The quadratic formula is a powerful tool for solving quadratic equations, but it can be intimidating for those who are new to it. In this article, we will answer some of the most frequently asked questions about the quadratic formula, covering topics such as its derivation, how to use it, and common mistakes to avoid.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations of the form $ax^2 + bx + c = 0$. 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 identify the coefficients $a$, $b$, and $c$ in the quadratic equation. Then, you substitute these values into the formula and simplify the expression under the square root. Finally, you solve for $x$.

Q: What is the significance of the square root in the quadratic formula?

A: The square root in the quadratic formula represents the discriminant, which is the expression under the square root. The discriminant determines the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: How do I simplify the expression under the square root?

A: To simplify the expression under the square root, you need to factor the expression into its prime factors. Then, you can take the square root of each factor and simplify the expression.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Not identifying the coefficients $a$, $b$, and $c$ correctly
• Not simplifying the expression under the square root correctly
• Not solving for $x$ correctly
• Not checking the nature of the roots correctly

Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you need to be careful when simplifying the expression under the square root, as it may involve complex numbers.

Q: Can I use the quadratic formula to solve quadratic equations with rational coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with rational coefficients. However, you need to be careful when simplifying the expression under the square root, as it may involve rational numbers.

Q: Is there a way to simplify the quadratic formula?

A: Yes, there are several ways to simplify the quadratic formula. One way is to use the fact that the quadratic formula can be written in the form:

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

This can be further simplified by factoring the expression under the square root.

Conclusion

In this article, we have answered some of the most frequently asked questions about the quadratic formula, covering topics such as its derivation, how to use it, and common mistakes to avoid. We hope that this article has provided a clear and concise explanation of the quadratic formula and its applications.

Additional Resources

For more information on the quadratic formula, we recommend the following resources:

• Khan Academy: Quadratic Formula
• Mathway: Quadratic Formula
• Wolfram Alpha: Quadratic Formula

These resources provide a comprehensive overview of the quadratic formula and its applications, as well as interactive tools and examples to help you practice and master the formula.