Solve The Equation:${ \begin{array}{c} 3x^2 + 4x - 3 = 0 \ x = \underline{[?] \pm \sqrt{\square}} \end{array} }$

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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 this article, we will focus on solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. We will use the quadratic formula to solve the equation 3x2+4xβˆ’3=03x^2 + 4x - 3 = 0 and provide a step-by-step guide on how to do it.

The Quadratic Formula

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

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation.

Solving the Equation 3x2+4xβˆ’3=03x^2 + 4x - 3 = 0

To solve the equation 3x2+4xβˆ’3=03x^2 + 4x - 3 = 0, we will use the quadratic formula. We have:

a=3a = 3, b=4b = 4, and c=βˆ’3c = -3

Plugging these values into the quadratic formula, we get:

x=βˆ’4Β±42βˆ’4(3)(βˆ’3)2(3)x = \frac{-4 \pm \sqrt{4^2 - 4(3)(-3)}}{2(3)}

Simplifying the Expression

To simplify the expression, we will start by evaluating the expression inside the square root:

42βˆ’4(3)(βˆ’3)=16+36=524^2 - 4(3)(-3) = 16 + 36 = 52

So, the expression becomes:

x=βˆ’4Β±526x = \frac{-4 \pm \sqrt{52}}{6}

Simplifying the Square Root

The square root of 52 can be simplified as:

52=4β‹…13=213\sqrt{52} = \sqrt{4 \cdot 13} = 2\sqrt{13}

So, the expression becomes:

x=βˆ’4Β±2136x = \frac{-4 \pm 2\sqrt{13}}{6}

Simplifying the Expression Further

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

x=βˆ’2Β±133x = \frac{-2 \pm \sqrt{13}}{3}

Conclusion

In this article, we solved the quadratic equation 3x2+4xβˆ’3=03x^2 + 4x - 3 = 0 using the quadratic formula. We started by plugging the values of aa, bb, and cc into the formula and then simplified the expression to get the final solution. The solution is given by:

x=βˆ’2Β±133x = \frac{-2 \pm \sqrt{13}}{3}

This is a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0, where a=3a = 3, b=4b = 4, and c=βˆ’3c = -3. The quadratic formula is a powerful tool for solving quadratic equations, and it is essential to understand how to use it to solve equations of this form.

Tips and Tricks

  • When using the quadratic formula, make sure to plug in the correct values of aa, bb, and cc.
  • Simplify the expression inside the square root before plugging it into the formula.
  • Divide both the numerator and the denominator by the greatest common divisor to simplify the expression further.

Real-World Applications

Quadratic equations have many real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Common Mistakes

  • Failing to simplify the expression inside the square root.
  • Failing to divide both the numerator and the denominator by the greatest common divisor.
  • Plugging in the wrong values of aa, bb, and cc into the formula.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and it is essential to understand how to use it to solve equations of this form. By following the steps outlined in this article, you can solve quadratic equations with ease and apply them to real-world problems.

Final Answer

The final answer is:

x = \frac{-2 \pm \sqrt{13}}{3}$<br/> # **Quadratic Equations Q&A: Frequently Asked Questions** =====================================================

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 answer some of the most frequently asked questions about quadratic equations, including how to solve them, common mistakes to avoid, and real-world applications.

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: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

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

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions.

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

A: Some common mistakes to avoid when solving quadratic equations include:

  • Failing to simplify the expression inside the square root
  • Failing to divide both the numerator and the denominator by the greatest common divisor
  • Plugging in the wrong values of a, b, and c into the formula

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation. However, it is essential to plug in the correct values of a, b, and c into the formula.

Q: What is the significance of the discriminant (b^2 - 4ac) in the quadratic formula?

A: The discriminant (b^2 - 4ac) is a crucial part of the quadratic formula. It determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

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

A: Yes, the quadratic formula can be used to solve quadratic equations with complex solutions. However, it is essential to remember that complex solutions are typically expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Q: What are some tips for simplifying the expression inside the square root in the quadratic formula?

A: Some tips for simplifying the expression inside the square root in the quadratic formula include:

  • Factoring the expression inside the square root
  • Using the difference of squares formula
  • Simplifying the expression by combining like terms

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the quadratic formula and avoiding common mistakes, you can solve quadratic equations with ease and apply them to real-world problems.

Final Answer

The final answer is:

x = (-b ± √(b^2 - 4ac)) / 2a