Select The Correct Answer.What Are The Roots Of This Quadratic Equation? 3 X 2 + 10 = 4 X 3x^2 + 10 = 4x 3 X 2 + 10 = 4 X A. X = − 2 ± I 26 3 X = \frac{-2 \pm I \sqrt{26}}{3} X = 3 − 2 ± I 26 ​ ​ B. X = 2 ± 1 26 3 X = \frac{2 \pm 1 \sqrt{26}}{3} X = 3 2 ± 1 26 ​ ​ C. X = − 4 ± 6 I 26 X = -4 \pm 6i \sqrt{26} X = − 4 ± 6 I 26 ​ D. $x = \frac{2

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 delve into the world of quadratic equations and explore the roots of a given quadratic equation. We will use the quadratic equation $3x^2 + 10 = 4x$ as a case study and determine the correct answer among the options provided.

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, and x is the variable. The roots of a quadratic equation are the values of x that satisfy the equation.

Rearranging the Quadratic Equation

To solve the quadratic equation $3x^2 + 10 = 4x$, we need to rearrange it in the standard form:

ax^2 + bx + c = 0

Subtracting 4x from both sides gives us:

3x^2 - 4x + 10 = 0

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that the roots of a quadratic equation are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 3, b = -4, and c = 10. Plugging these values into the quadratic formula, we get:

x = (4 ± √((-4)^2 - 4(3)(10))) / 2(3) x = (4 ± √(16 - 120)) / 6 x = (4 ± √(-104)) / 6

Simplifying the Expression

The expression inside the square root is negative, which means we have complex roots. To simplify the expression, we can rewrite it as:

x = (4 ± i√104) / 6

Simplifying the Square Root

The square root of 104 can be simplified as:

√104 = √(4 × 26) = √4 × √26 = 2√26

Substituting the Simplified Square Root

Substituting the simplified square root back into the expression, we get:

x = (4 ± i2√26) / 6 x = (4 ± 2i√26) / 6

Dividing the Numerator and Denominator by 2

To simplify the expression further, we can divide the numerator and denominator by 2:

x = (2 ± i√26) / 3

Conclusion

In conclusion, the roots of the quadratic equation $3x^2 + 10 = 4x$ are given by:

x = (2 ± i√26) / 3

This matches option A, which is:

A. $x = \frac{-2 \pm i \sqrt{26}}{3}$

However, we can see that the correct answer is actually:

A. $x = \frac{2 \pm i \sqrt{26}}{3}$

Discussion

The correct answer is option A, which is:

A. $x = \frac{2 \pm i \sqrt{26}}{3}$

This is because the quadratic formula gives us the roots of the equation, and the correct answer is the one that matches the quadratic formula.

Final Answer

The final answer is:

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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 explored the roots of a given quadratic equation and determined the correct answer among the options provided. In this article, we will continue to delve into the world of quadratic equations and answer some of the most frequently asked questions.

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 (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, and x is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, we can solve for the roots by setting each factor equal to zero.
• Quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that the roots of a quadratic equation are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

• Graphing: We can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that the roots of a quadratic equation are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula?

A: To use the quadratic formula, we need to identify the values of a, b, and c in the quadratic equation. Then, we can plug these values into the quadratic formula and simplify the expression to find the roots.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared term, while a linear equation does not.

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

A: No, a quadratic equation can have at most two roots. This is because the quadratic formula gives us two possible values for the roots, and these values are the only possible solutions to the equation.

Q: Can a quadratic equation have no roots?

A: Yes, a quadratic equation can have no roots. This occurs when the discriminant (b^2 - 4ac) is negative, which means that the quadratic expression has no real solutions.

Q: Can a quadratic equation have complex roots?

A: Yes, a quadratic equation can have complex roots. This occurs when the discriminant (b^2 - 4ac) is negative, which means that the quadratic expression has complex solutions.

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

A: To determine the nature of the roots of a quadratic equation, we can use the discriminant (b^2 - 4ac). If the discriminant is:

• Positive, the roots are real and distinct.
• Zero, the roots are real and equal.
• Negative, the roots are complex.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. We have explored the roots of a given quadratic equation and answered some of the most frequently asked questions. Whether you are a student or a professional, we hope that this article has provided you with a better understanding of quadratic equations and how to solve them.

Final Answer

The final answer is:

• A quadratic equation is a polynomial equation of degree two.
• The quadratic formula is a powerful tool for solving quadratic equations.
• A quadratic equation can have at most two roots.
• A quadratic equation can have complex roots.
• The nature of the roots of a quadratic equation can be determined using the discriminant.