Solve For $x$ Using The Quadratic Formula.${ 2x^2 + 14x + 12 = 0 }$ X = ? X = \, ? X = ? $

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Introduction

The quadratic formula is a powerful tool used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation has real or complex roots. In this article, we will use the quadratic formula to solve the equation 2x2+14x+12=02x^2 + 14x + 12 = 0.

The Quadratic Formula

The quadratic formula 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. In our example, a=2a = 2, b=14b = 14, and c=12c = 12.

Step 1: Plug in the Values

To use the quadratic formula, we need to plug in the values of aa, bb, and cc into the formula. We will start by plugging in the values of aa and bb:

x=−14±142−4(2)(12)2(2)x = \frac{-14 \pm \sqrt{14^2 - 4(2)(12)}}{2(2)}

Step 2: Simplify the Expression

Next, we need to simplify the expression inside the square root. We will start by evaluating the expression 142−4(2)(12)14^2 - 4(2)(12):

142−4(2)(12)=196−96=10014^2 - 4(2)(12) = 196 - 96 = 100

Step 3: Simplify the Quadratic Formula

Now that we have simplified the expression inside the square root, we can simplify the quadratic formula:

x=−14±1004x = \frac{-14 \pm \sqrt{100}}{4}

Step 4: Simplify the Square Root

The square root of 100 is 10, so we can simplify the quadratic formula further:

x=−14±104x = \frac{-14 \pm 10}{4}

Step 5: Solve for xx

Finally, we can solve for xx by evaluating the two possible solutions:

x=−14+104=−44=−1x = \frac{-14 + 10}{4} = \frac{-4}{4} = -1

x=−14−104=−244=−6x = \frac{-14 - 10}{4} = \frac{-24}{4} = -6

Conclusion

In this article, we used the quadratic formula to solve the equation 2x2+14x+12=02x^2 + 14x + 12 = 0. We plugged in the values of aa, bb, and cc into the formula, simplified the expression inside the square root, and finally solved for xx. The two possible solutions are x=−1x = -1 and x=−6x = -6.

Why is the Quadratic Formula Important?

The quadratic formula is an important tool in mathematics because it can be used to solve quadratic equations of any degree. It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation has real or complex roots. The quadratic formula is used in a wide range of applications, including physics, engineering, and computer science.

Real- Applications of the Quadratic Formula

The quadratic formula has many real-world applications. For example, it can be used to model the motion of objects under the influence of gravity, to determine the trajectory of a projectile, and to calculate the stress on a beam. It can also be used to solve problems in physics, engineering, and computer science.

How to Use the Quadratic Formula in Real-World Applications

To use the quadratic formula in real-world applications, you need to have a good understanding of the formula and how to apply it. You also need to have a good understanding of the problem you are trying to solve and how to translate it into a quadratic equation. Once you have done this, you can use the quadratic formula to find the solutions to the equation.

Tips and Tricks for Using the Quadratic Formula

Here are some tips and tricks for using the quadratic formula:

  • Make sure to plug in the values of aa, bb, and cc correctly into the formula.
  • Simplify the expression inside the square root before plugging it into the formula.
  • Use the quadratic formula to solve quadratic equations of any degree.
  • Use the quadratic formula to find the solutions to any quadratic equation, regardless of whether the equation has real or complex roots.

Conclusion

In conclusion, the quadratic formula is a powerful tool used to solve quadratic equations of any degree. It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation has real or complex roots. The quadratic formula has many real-world applications and is used in a wide range of fields, including physics, engineering, and computer science. By following the steps outlined in this article, you can use the quadratic formula to solve quadratic equations and apply it to real-world problems.

Final Answer

The final answer is −1\boxed{-1} and −6\boxed{-6}.

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Introduction

The quadratic formula is a powerful tool used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. In our previous article, we used the quadratic formula to solve the equation 2x2+14x+12=02x^2 + 14x + 12 = 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 general method used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is given by:

x=−b±b2−4ac2ax = \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 plug in the values of aa, bb, and cc into the formula. You also need to simplify the expression inside the square root before plugging it into the formula.

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

A: The quadratic formula and factoring are two different methods used to solve 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 to the equation.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex roots. The quadratic formula will give you two solutions, one of which will be a complex number.

Q: How do I determine whether a quadratic equation has real or complex roots?

A: To determine whether a quadratic equation has real or complex roots, you need to look at the discriminant, which is the expression inside the square root in the quadratic formula. If the discriminant is positive, the equation has two real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with rational coefficients. The quadratic formula will give you two solutions, one of which will be a rational number.

Q: How do I simplify the expression inside the square root in the quadratic formula?

A: To simplify the expression inside the square root in the quadratic formula, you need to evaluate the expression b2−4acb^2 - 4ac. If the result is a perfect square, you can simplify the expression further.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with negative coefficients. The quadratic formula will give you two solutions, one of which will be a negative number.

Q: How do I apply the quadratic formula to real-world problems?

A: To apply the quadratic formula to real-world problems, you need to have a good understanding of the formula and how to apply it. You also need to have a good understanding of the problem you are trying to solve and how to translate it into a quadratic equation.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

  • Plugging in the values of aa, bb, and cc incorrectly into the formula.
  • Failing to simplify the expression inside the square root before plugging it into the formula.
  • Using the quadratic formula to solve quadratic equations that are not in the form ax2+bx+c=0ax^2 + bx + c = 0.

Conclusion

In conclusion, the quadratic formula is a powerful tool used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation has real or complex roots. By following the steps outlined in this article, you can use the quadratic formula to solve quadratic equations and apply it to real-world problems.

Final Answer

The final answer is that the quadratic formula is a powerful tool used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation has real or complex roots.