Use The Quadratic Formula To Find The Solutions To The Equation. 3 X 2 − 10 X + 5 = 0 3x^2 - 10x + 5 = 0 3 X 2 − 10 X + 5 = 0 A. 10 ± 40 6 \frac{10 \pm \sqrt{40}}{6} 6 10 ± 40 ​ ​ B. 2 ± 24 2 \frac{2 \pm \sqrt{24}}{2} 2 2 ± 24 ​ ​ C. 1 ± 35 2 \frac{1 \pm \sqrt{35}}{2} 2 1 ± 35 ​ ​ D. 5 ± 15 3 \frac{5 \pm \sqrt{15}}{3} 3 5 ± 15 ​ ​

Introduction to the Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions to quadratic equations of the form $ax^2 + bx + c = 0$. It is a fundamental concept in algebra and is widely used in various fields such as physics, engineering, and economics. In this article, we will explore how to use the quadratic formula to find the solutions to the equation $3x^2 - 10x + 5 = 0$.

Understanding the Quadratic Formula

The quadratic 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. To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the given equation.

Identifying the Coefficients

In the equation $3x^2 - 10x + 5 = 0$, we can identify the coefficients as follows:

• $a = 3$
• $b = -10$
• $c = 5$

Applying the Quadratic Formula

Now that we have identified the coefficients, we can apply the quadratic formula to find the solutions to the equation. Plugging in the values of $a$, $b$, and $c$ into the quadratic formula, we get:

$$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(5)}}{2(3)}$$

Simplifying the expression, we get:

$$x = \frac{10 \pm \sqrt{100 - 60}}{6}$$

$$x = \frac{10 \pm \sqrt{40}}{6}$$

Evaluating the Solutions

The quadratic formula gives us two possible solutions for the equation:

$$x = \frac{10 + \sqrt{40}}{6}$$

$$x = \frac{10 - \sqrt{40}}{6}$$

We can simplify the solutions further by evaluating the square root of 40:

$$\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$$

Substituting this value back into the solutions, we get:

$$x = \frac{10 + 2\sqrt{10}}{6}$$

$$x = \frac{10 - 2\sqrt{10}}{6}$$

Comparing the Solutions

Now that we have evaluated the solutions, we can compare them to the answer choices provided:

A. $\frac{10 \pm \sqrt{40}}{6}$

B. $\frac{2 \pm \sqrt{24}}{2}$

C. $\frac{1 \pm \sqrt{35}}{2}$

D. $\frac{5 \pm \sqrt{15}}{3}$

We can see that the solutions we obtained match option A.

Conclusion

In this article, we used the quadratic formula to find the solutions to the equation $3x^2 - 10x + 5 = 0$. We identified the coefficients of the equation, applied the quadratic formula, and evaluated the solutions. We compared the solutions to the answer choices provided and found that the correct solution is option A.

Final Answer

The final answer is $\boxed{\frac{10 \pm \sqrt{40}}{6}}$.

Introduction

The quadratic formula is a powerful tool used to find the solutions to quadratic equations of the form $ax^2 + bx + c = 0$. In our previous article, we explored how to use the quadratic formula to find the solutions to the equation $3x^2 - 10x + 5 = 0$. In this article, we will answer some frequently asked questions about the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to find the solutions to 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 of the quadratic equation, which are $a$, $b$, and $c$. Then, you plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What are the coefficients of the quadratic equation?

A: The coefficients of the quadratic equation are $a$, $b$, and $c$. In the equation $ax^2 + bx + c = 0$, $a$ is the coefficient of the $x^2$ term, $b$ is the coefficient of the $x$ term, and $c$ is the constant term.

Q: How do I simplify the quadratic formula?

A: To simplify the quadratic formula, you need to evaluate the expression inside the square root, which is $b^2 - 4ac$. Then, you can simplify the expression further by factoring out any common factors.

Q: What are the solutions to the quadratic equation?

A: The solutions to the quadratic equation are the values of $x$ that satisfy the equation. They are given by the quadratic formula:

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

Q: How do I choose the correct solution?

A: To choose the correct solution, you need to evaluate the expression inside the square root, which is $b^2 - 4ac$. If the expression is positive, then the solutions are real numbers. If the expression is negative, then the solutions are complex numbers.

Q: What are the applications of the quadratic formula?

A: The quadratic formula has many applications in various fields such as physics, engineering, and economics. It is used to model real-world problems such as projectile motion, electrical circuits, and population growth.

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

A: No, the quadratic formula is only used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It cannot be used to solve cubic equations or higher-degree equations.

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

A: Yes, the quadratic formula can be used to solve equations with complex coefficients. However, the solutions may be complex numbers.

Conclusion

In this article, we answered some frequently asked questions about the quadratic formula. We covered topics such as the definition of the quadratic formula, how to use it, and its applications. We also discussed some common mistakes to avoid when using the quadratic formula.

Final Answer

The final answer is that the quadratic formula is a powerful tool used to find the solutions to quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

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

We hope that this article has been helpful in answering your questions about the quadratic formula.