What Are The Solution(s) Of The Quadratic Equation 6 X 2 − 1 X − 12 = 0 6x^2 - 1x - 12 = 0 6 X 2 − 1 X − 12 = 0 ?

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications 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 ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable. In this article, we will focus on solving the quadratic equation 6x2−1x−12=06x^2 - 1x - 12 = 0.

The Quadratic Formula

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

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation. To use the quadratic formula, we need to identify the values of aa, bb, and cc in the given equation.

Identifying the Coefficients

In the equation 6x2−1x−12=06x^2 - 1x - 12 = 0, we can identify the coefficients as follows:

a=6a = 6 b=−1b = -1 c=−12c = -12

Applying the Quadratic Formula

Now that we have identified the coefficients, we can apply the quadratic formula to solve the equation. Plugging in the values of aa, bb, and cc into the formula, we get:

x=−(−1)±(−1)2−4(6)(−12)2(6)x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(6)(-12)}}{2(6)}

Simplifying the expression, we get:

x=1±1+28812x = \frac{1 \pm \sqrt{1 + 288}}{12}

x=1±28912x = \frac{1 \pm \sqrt{289}}{12}

x=1±1712x = \frac{1 \pm 17}{12}

Solving for x

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

Case 1: x=1+1712x = \frac{1 + 17}{12}

x=1812x = \frac{18}{12}

x=32x = \frac{3}{2}

Case 2: x=1−1712x = \frac{1 - 17}{12}

x=−1612x = \frac{-16}{12}

x=−43x = -\frac{4}{3}

Conclusion

In this article, we have solved the quadratic equation 6x2−1x−12=06x^2 - 1x - 12 = 0 using the quadratic formula. We identified the coefficients of the equation, applied the quadratic formula, and solved for xx by considering the two possible cases. The solutions to the equation are x=32x = \frac{3}{2} and x=−43x = -\frac{4}{3}. These solutions can be verified by plugging them back into the original equation.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields such as physics, engineering, and economics. Some of the applications of quadratic equations include:

  • Projectile Motion: Quadratic equations are used to model the trajectory of projectiles under the influence of gravity.
  • Optimization: Quadratic equations are used to optimize functions and find the maximum or minimum value of a function.
  • Electrical Engineering: Quadratic equations are used to analyze and design electrical circuits.
  • Computer Graphics: Quadratic equations are used to create 3D models and animations.

Tips for Solving Quadratic Equations

Solving quadratic equations can be challenging, but there are some tips that can make it easier:

  • Use the Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations.
  • Identify the Coefficients: Identifying the coefficients of the equation is crucial for applying the quadratic formula.
  • Simplify the Expression: Simplifying the expression can make it easier to solve for xx.
  • Consider the Two Possible Cases: Considering the two possible cases can help you find the solutions to the equation.

Conclusion

In conclusion, solving quadratic equations is an essential skill in mathematics, and it has numerous applications in various fields. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions to the equation. By identifying the coefficients, simplifying the expression, and considering the two possible cases, we can solve quadratic equations and find the solutions to the equation.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. In our previous article, we solved the quadratic equation 6x2−1x−12=06x^2 - 1x - 12 = 0 using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations.

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 is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx 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 equation can be factored, we can solve it by finding the factors.
  • Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{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 is given by: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the quadratic equation.

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, we need to identify the values of aa, bb, and cc in the given equation. Then, we can plug these values into the formula and simplify the expression to find the solutions.

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of xx that satisfy the equation. In other words, they are the values of xx that make the equation true.

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

A: No, a quadratic equation can have at most two solutions. This is because the graph of a quadratic function is a parabola, which has at most two x-intercepts.

Q: How do I determine the number of solutions to a quadratic equation?

A: We can determine the number of solutions to a quadratic equation by examining the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It is given by: b2−4acb^2 - 4ac.

Q: How do I use the discriminant to determine the number of solutions?

A: We can use the discriminant to determine the number of solutions to a quadratic equation as follows:

  • Positive discriminant: If the discriminant is positive, the equation has two distinct solutions.
  • Zero discriminant: If the discriminant is zero, the equation has one repeated solution.
  • Negative discriminant: If the discriminant is negative, the equation has no real solutions.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. This occurs when the discriminant is negative.

Q: How do I find the complex solutions to a quadratic equation?

A: To find the complex solutions to a quadratic equation, we can use the quadratic formula and take the square root of the negative discriminant.

Q: What are some common applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields such as physics, engineering, and economics. Some of the common applications of quadratic equations include:

  • Projectile Motion: Quadratic equations are used to model the trajectory of projectiles under the influence of gravity.
  • Optimization: Quadratic equations are used to optimize functions and find the maximum or minimum value of a function.
  • Electrical Engineering: Quadratic equations are used to analyze and design electrical circuits.
  • Computer Graphics: Quadratic equations are used to create 3D models and animations.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the quadratic formula and the discriminant, we can solve quadratic equations and determine the number of solutions. We can also use quadratic equations to model real-world phenomena and optimize functions.