Solve The Equation Using The Quadratic Formula: 15 X 2 + 13 X = 0 15x^2 + 13x = 0 15 X 2 + 13 X = 0 Please Select The Best Answer From The Choices Provided:A. X = − 13 15 , 0 X = -\frac{13}{15}, 0 X = − 15 13 ​ , 0 B. X = 0 X = 0 X = 0 C. X = 13 15 , 0 X = \frac{13}{15}, 0 X = 15 13 ​ , 0 D. $x = \pm

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 quadratic equations using the quadratic formula, a powerful tool for finding the solutions to quadratic equations. We will use the equation $15x^2 + 13x = 0$ as an example to demonstrate the steps involved in solving quadratic equations using the quadratic formula.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Step 1: Identify the Coefficients

To use the quadratic formula, we need to identify the coefficients $a$, $b$, and $c$ in the given quadratic equation. In the equation $15x^2 + 13x = 0$, we can see that:

• $a = 15$
• $b = 13$
• $c = 0$

Step 2: Plug in the Values

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

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

Substituting the values of $a$, $b$, and $c$, we get:

$x = \frac{-13 \pm \sqrt{13^2 - 4(15)(0)}}{2(15)}$

Step 3: Simplify the Expression

Simplifying the expression inside the square root, we get:

$x = \frac{-13 \pm \sqrt{169}}{30}$

Since $\sqrt{169} = 13$, we can further simplify the expression:

$x = \frac{-13 \pm 13}{30}$

Step 4: Solve for x

Now that we have simplified the expression, we can solve for $x$ by considering the two possible cases:

Case 1: $x = \frac{-13 + 13}{30}$ $x = \frac{0}{30}$ $x = 0$

Case 2: $x = \frac{-13 - 13}{30}$ $x = \frac{-26}{30}$ $x = -\frac{13}{15}$

Conclusion

In this article, we have demonstrated how to solve quadratic equations using the quadratic formula. We used the equation $15x^2 + 13x = 0$ as an example and followed the steps involved in solving quadratic equations using the quadratic formula. We identified the coefficients, plugged them into the quadratic formula, simplified the expression, and solved for $x$. The solutions to the equation are $x = 0$ and $x = -\frac{13}{15}$.

Answer

Based on the steps outlined above, the correct answer is:

A. $x = -\frac{13}{15}, 0$

This answer is consistent with the solutions obtained using the quadratic formula.

Final Thoughts

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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 our previous article, we demonstrated how to solve quadratic equations using the quadratic formula. In this article, we will provide a Q&A guide to help you better understand quadratic equations and how to solve them.

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 (usually x) is two. It is typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to quadratic equations of the form ax^2 + bx + c = 0. It is given by:

x = (-b ± √(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 given quadratic equation. Then, plug them into the quadratic formula and simplify the expression to find the solutions.

Q: What are the steps involved in solving quadratic equations using the quadratic formula?

A: The steps involved in solving quadratic equations using the quadratic formula are:

1. Identify the coefficients a, b, and c in the given quadratic equation.
2. Plug them into the quadratic formula.
3. Simplify the expression inside the square root.
4. Solve for x by considering the two possible cases.

Q: What are the two possible cases when solving quadratic equations using the quadratic formula?

A: When solving quadratic equations using the quadratic formula, you need to consider two possible cases:

Case 1: x = (-b + √(b^2 - 4ac)) / 2a Case 2: x = (-b - √(b^2 - 4ac)) / 2a

Q: What happens if the expression inside the square root is negative?

A: If the expression inside the square root is negative, then the quadratic equation has no real solutions. In this case, the solutions are complex numbers.

Q: How do I determine if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to check the expression inside the square root. If it is positive, then the equation has real solutions. If it is negative, then the equation has complex solutions.

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

A: Some common mistakes to avoid when solving quadratic equations using the quadratic formula are:

• Not identifying the coefficients a, b, and c correctly.
• Not simplifying the expression inside the square root correctly.
• Not considering the two possible cases when solving for x.

Conclusion

In this article, we have provided a Q&A guide to help you better understand quadratic equations and how to solve them using the quadratic formula. We have covered topics such as the definition of a quadratic equation, the quadratic formula, and the steps involved in solving quadratic equations using the quadratic formula. We have also discussed common mistakes to avoid when solving quadratic equations using the quadratic formula.

Final Thoughts

Solving quadratic equations using the quadratic formula is a powerful tool for finding the solutions to quadratic equations. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving quadratic equations using the quadratic formula.