What Are The Solutions Of $8x^2 = 6 + 22x$?Check All Of The Boxes That Apply.- X = − 3 X = -3 X = − 3 - X = − 1 4 X = -\frac{1}{4} X = − 4 1 ​ - X = 3 X = 3 X = 3 - X = 6 X = 6 X = 6

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods and techniques used to find the solutions. In this article, we will focus on solving the quadratic equation $8x^2 = 6 + 22x$ and check which of the given options are correct solutions.

Understanding the 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 aa, bb, and cc are constants. In our given equation, $8x^2 = 6 + 22x$, we can rewrite it in the standard form as $8x^2 - 22x - 6 = 0$.

Rearranging the Equation

To solve the equation, we need to rearrange it in the standard form. We can do this by subtracting 66 from both sides of the equation and then subtracting 22x22x from both sides. This gives us $8x^2 - 22x - 6 = 0$.

Factoring the Equation

One of the methods to solve a quadratic equation is by factoring. However, in this case, the equation does not factor easily. Therefore, we will use the quadratic formula to find the solutions.

Using the Quadratic Formula

The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our equation, a=8a = 8, b=−22b = -22, and c=−6c = -6. Plugging these values into the formula, we get $x = \frac{-(-22) \pm \sqrt{(-22)^2 - 4(8)(-6)}}{2(8)}$.

Simplifying the Expression

Simplifying the expression inside the square root, we get $x = \frac{22 \pm \sqrt{484 + 192}}{16}$.

Continuing the Simplification

Continuing the simplification, we get $x = \frac{22 \pm \sqrt{676}}{16}$.

Final Simplification

The square root of 676676 is 2626. Therefore, the final simplification is $x = \frac{22 \pm 26}{16}$.

Finding the Solutions

Now, we can find the two solutions by plugging in the plus and minus signs separately. For the plus sign, we get $x = \frac{22 + 26}{16} = \frac{48}{16} = 3$. For the minus sign, we get $x = \frac{22 - 26}{16} = \frac{-4}{16} = -\frac{1}{4}$.

Checking the Solutions

Now that we have found the solutions, we need to check which of the given options are correct. The options are x=−3x = -3, x=−14x = -\frac{1}{4}, x=3x = 3, and x=6x = 6. We can plug in each of these values into the original equation to check if they are true solutions.

Checking x=−3x = -3

Plugging in x=−3x = -3 into the original equation, we get $8(-3)^2 = 6 + 22(-3)$. Simplifying this expression, we get $72 = -66$. This is not true, so x=−3x = -3 is not a solution.

Checking x=−14x = -\frac{1}{4}

Plugging in x=−14x = -\frac{1}{4} into the original equation, we get $8(-\frac{1}{4})^2 = 6 + 22(-\frac{1}{4})$. Simplifying this expression, we get $\frac{1}{2} = -\frac{11}{2}$. This is not true, so x=−14x = -\frac{1}{4} is not a solution.

Checking x=3x = 3

Plugging in x=3x = 3 into the original equation, we get $8(3)^2 = 6 + 22(3)$. Simplifying this expression, we get $72 = 72$. This is true, so x=3x = 3 is a solution.

Checking x=6x = 6

Plugging in x=6x = 6 into the original equation, we get $8(6)^2 = 6 + 22(6)$. Simplifying this expression, we get $432 = 138$. This is not true, so x=6x = 6 is not a solution.

Conclusion

In this article, we solved the quadratic equation $8x^2 = 6 + 22x$ using the quadratic formula. We found the two solutions to be x=3x = 3 and x=−14x = -\frac{1}{4}. We then checked each of the given options to see if they were true solutions. The only correct solution was x=3x = 3.

Final Answer

The final answer is: 3\boxed{3}

Introduction

In our previous article, we solved the quadratic equation $8x^2 = 6 + 22x$ and found the solutions to be x=3x = 3 and x=−14x = -\frac{1}{4}. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to solve 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 $ax^2 + bx + c = 0$, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and the type of solutions you are looking for.

Q: What is the quadratic formula?

A: The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can be used to find the solutions to a quadratic equation when it cannot be factored easily.

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 will then get two solutions, which are the values of xx that satisfy the equation.

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

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

  1. Plug in the values of aa, bb, and cc into the formula.
  2. Simplify the expression inside the square root.
  3. Simplify the expression outside the square root.
  4. Find the two solutions by plugging in the plus and minus signs separately.

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

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

  • Not simplifying the expression inside the square root.
  • Not simplifying the expression outside the square root.
  • Not finding the two solutions by plugging in the plus and minus signs separately.
  • Not checking the solutions to see if they are true solutions.

Q: How do I check if a solution is a true solution?

A: To check if a solution is a true solution, you need to plug it back into the original equation and see if it is true. If the solution satisfies the equation, then it is a true solution.

Q: What are the different types of solutions to a quadratic equation?

A: The different types of solutions to a quadratic equation include:

  • Real solutions: These are solutions that are real numbers.
  • Complex solutions: These are solutions that are complex numbers.
  • Rational solutions: These are solutions that are rational numbers.
  • Irrational solutions: These are solutions that are irrational numbers.

Q: How do I determine the type of solution to a quadratic equation?

A: To determine the type of solution to a quadratic equation, you need to look at the discriminant, which is the expression inside the square root in the quadratic formula. If the discriminant is positive, then the solutions are real. If the discriminant is negative, then the solutions are complex.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques used to solve quadratic equations. We covered topics such as the quadratic formula, how to use it, and how to check if a solution is a true solution. We also discussed the different types of solutions to a quadratic equation and how to determine the type of solution.

Final Answer

The final answer is: 3\boxed{3}