Using The Quadratic Formula, Which Of The Following Are The Zeros Of The Quadratic Equation Below? Y = 3 X 2 − 10 X + 5 Y=3x^2-10x+5 Y = 3 X 2 − 10 X + 5 A. X = − 5 ± 50 3 X=\frac{-5 \pm \sqrt{50}}{3} X = 3 − 5 ± 50 B. X = − 5 ± 10 3 X=\frac{-5 \pm \sqrt{10}}{3} X = 3 − 5 ± 10 C. X = 5 ± 50 3 X=\frac{5 \pm \sqrt{50}}{3} X = 3 5 ± 50

Mar 10, 2025 by ADMIN 347 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

In this article, we will focus on solving quadratic equations using the quadratic formula. The quadratic formula is a powerful tool that allows us to find the solutions of a quadratic equation in the form of $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . We will use this formula to solve the quadratic equation $y = 3x^2 - 10x + 5$ and determine which of the given options are the zeros of the equation.

The Quadratic Formula

The quadratic formula is a fundamental concept in mathematics, and it is used to solve quadratic equations. 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, and $x$ is the variable.

To use the quadratic formula, we need to identify the values of $a$ , $b$ , and $c$ in the given quadratic equation. In the equation $y = 3x^2 - 10x + 5$ , we have:

$a = 3$ $b = -10$ $c = 5$

Now, we can plug these values into the quadratic formula to find the solutions of the equation.

Solving the Quadratic Equation

Using the quadratic formula, we get:

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

Simplifying the expression, we get:

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

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

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

$x = \frac{5 \pm \sqrt{10}}{3}$

Therefore, the solutions of the quadratic equation $y = 3x^2 - 10x + 5$ are $x = \frac{5 + \sqrt{10}}{3}$ and $x = \frac{5 - \sqrt{10}}{3}$ .

Comparing the Solutions

Now, let's compare the solutions we obtained with the given options.

Option A: $x = \frac{-5 \pm \sqrt{50}}{3}$

Option B: $x = \frac{-5 \pm \sqrt{10}}{3}$

Option C: $x = \frac{5 \pm \sqrt{50}}{3}$

We can see that option B matches our solution, which is $x = \frac{5 \pm \sqrt{10}}{3}$ .

Conclusion

In this article, we used the quadratic formula to solve the quadratic equation $y = 3x^2 - 10x + 5$ . We obtained the solutions $x = \frac{5 + \sqrt{10}}{3}$ and $x = \frac{5 - \sqrt{10}}{3}$ . We compared these solutions with the given options and found that option B matches our solution.

The quadratic formula is a powerful tool that allows us to solve quadratic equations. It is a fundamental concept in mathematics, and it has numerous applications in various fields. In this article, we demonstrated how to use the quadratic formula to solve a quadratic equation and determine the zeros of the equation.

Final Answer

The final answer is option B: $x = \frac{5 \pm \sqrt{10}}{3}$ .

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

Q1: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q2: What is the quadratic formula?

The quadratic formula is a powerful tool that allows us to find the solutions of a quadratic equation in the form of $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . This formula is used to solve quadratic equations when the solutions cannot be found by factoring.

Q3: How do I use the quadratic formula?

To use the quadratic formula, you need to identify the values of $a$ , $b$ , and $c$ in the given quadratic equation. Then, you can plug these values into the quadratic formula to find the solutions of the equation.

Q4: What are the steps to solve a quadratic equation using the quadratic formula?

The steps to solve a quadratic equation using the quadratic formula are:

Identify the values of $a$ , $b$ , and $c$ in the given quadratic equation.
Plug these values into the quadratic formula.
Simplify the expression to find the solutions of the equation.

Q5: What are the solutions of a quadratic equation?

The solutions of a quadratic equation are the values of $x$ that satisfy the equation. These solutions can be real or complex numbers.

Q6: How do I determine the zeros of a quadratic equation?

The zeros of a quadratic equation are the values of $x$ that make the equation equal to zero. To determine the zeros of a quadratic equation, you need to solve the equation using the quadratic formula.

Q7: What is the difference between a quadratic equation and a linear equation?

A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is $ax + b = 0$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q8: Can a quadratic equation have more than two solutions?

No, a quadratic equation can have at most two solutions. This is because the quadratic formula only provides two solutions for a quadratic equation.

Q9: Can a quadratic equation have no solutions?

Yes, a quadratic equation can have no solutions. This occurs when the discriminant ( $b^2 - 4ac$ ) is negative.

Q10: What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation is the expression $b^2 - 4ac$ . The discriminant determines the nature of the solutions of a quadratic equation.

Conclusion

In this article, we provided a Q&A guide to help you understand quadratic equations better. We discussed the definition of a quadratic equation, the quadratic formula, and the steps to solve a quadratic equation using the quadratic formula. We also answered questions about the solutions of a quadratic equation, the zeros of a quadratic equation, and the difference between a quadratic equation and a linear equation.

Final Answer

We hope this Q&A guide has helped you understand quadratic equations better. If you have any further questions, please feel free to ask.