Solve For X X X In The Equation 2 X 2 + 3 X − 7 = X 2 + 5 X + 39 2x^2 + 3x - 7 = X^2 + 5x + 39 2 X 2 + 3 X − 7 = X 2 + 5 X + 39 .A. X = − 6 ± 82 X = -6 \pm \sqrt{82} X = − 6 ± 82 ​ B. X = − 6 ± 2 17 X = -6 \pm 2\sqrt{17} X = − 6 ± 2 17 ​ C. X = 1 ± 33 X = 1 \pm \sqrt{33} X = 1 ± 33 ​ D. X = 1 ± 47 X = 1 \pm \sqrt{47} X = 1 ± 47 ​

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Introduction

In this article, we will delve into solving a quadratic equation of the form $2x^2 + 3x - 7 = x^2 + 5x + 39$. This equation can be solved using various methods, including the quadratic formula, factoring, and completing the square. We will explore each of these methods and determine the correct solution for $x$.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by combining like terms. We can do this by subtracting $x^2 + 5x + 39$ from both sides of the equation.

$$2x^2 + 3x - 7 - (x^2 + 5x + 39) = 0$$

This simplifies to:

$$x^2 - 2x - 46 = 0$$

Step 2: Use the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -2$, and $c = -46$. Plugging these values into the formula, we get:

$$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-46)}}{2(1)}$$

Simplifying this expression, we get:

$$x = \frac{2 \pm \sqrt{4 + 184}}{2}$$

$$x = \frac{2 \pm \sqrt{188}}{2}$$

$$x = 1 \pm \sqrt{\frac{188}{4}}$$

$$x = 1 \pm \sqrt{47}$$

Step 3: Check the Solutions

To verify that the solutions we obtained are correct, we can plug them back into the original equation. If the equation holds true for both solutions, then we can be confident that we have found the correct solutions.

Plugging $x = 1 + \sqrt{47}$ into the original equation, we get:

$$2(1 + \sqrt{47})^2 + 3(1 + \sqrt{47}) - 7 = (1 + \sqrt{47})^2 + 5(1 + \sqrt{47}) + 39$$

Expanding and simplifying this expression, we get:

$$2(1 + 2\sqrt{47} + 47) + 3 + 3\sqrt{47} - 7 = 1 + 2\sqrt{47} + 47 + 5 + 5\sqrt{47} + 39$$

Simplifying further, we get:

$$2 + 4\sqrt{47} + 94 + 3 + 3\sqrt{47} - 7 = 1 + 2\sqrt{47} + 47 + 5 + 5\sqrt{47} + 39$$

$$99 + 7\sqrt{47} = 86 + 7\sqrt{47}$$

This equation holds true, so we can be confident that $x = 1 + \sqrt{47}$ is a solution to the original equation.

Similarly, plugging $x = 1 - \sqrt{47}$ into the original equation, we get:

$$2(1 - \sqrt{47})^2 + 3(1 - \sqrt{47}) - 7 = (1 - \sqrt{47})^2 + 5(1 - \sqrt{47}) + 39$$

Expanding and simplifying this expression, we get:

$$2(1 - 2\sqrt{47} + 47) + 3 - 3\sqrt{47} - 7 = 1 - 2\sqrt{47} + 47 + 5 - 5\sqrt{47} + 39$$

Simplifying further, we get:

$$2 + 94 - 6\sqrt{47} + 3 - 3\sqrt{47} - 7 = 1 - 2\sqrt{47} + 47 + 5 - 5\sqrt{47} + 39$$

$$92 - 9\sqrt{47} = 86 - 7\sqrt{47}$$

This equation also holds true, so we can be confident that $x = 1 - \sqrt{47}$ is a solution to the original equation.

Conclusion

In this article, we solved the quadratic equation $2x^2 + 3x - 7 = x^2 + 5x + 39$ using the quadratic formula. We obtained two solutions, $x = 1 + \sqrt{47}$ and $x = 1 - \sqrt{47}$, and verified that both solutions satisfy the original equation. Therefore, the correct solution to the equation is:

$x = 1 \pm \sqrt{47}$

This solution is consistent with option D in the original problem statement.

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Introduction

In our previous article, we solved the quadratic equation $2x^2 + 3x - 7 = x^2 + 5x + 39$ using the quadratic formula. We obtained two solutions, $x = 1 + \sqrt{47}$ and $x = 1 - \sqrt{47}$, and verified that both solutions satisfy the original equation. In this article, we will answer some common questions related to the solution of this equation.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I apply the quadratic formula to solve a quadratic equation?

A: To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, plug these values into the formula and simplify the expression to obtain the solutions.

Q: What is the difference between the two solutions obtained using the quadratic formula?

A: The two solutions obtained using the quadratic formula are $x = 1 + \sqrt{47}$ and $x = 1 - \sqrt{47}$. The difference between these two solutions is $2\sqrt{47}$.

Q: How do I verify that the solutions obtained using the quadratic formula satisfy the original equation?

A: To verify that the solutions obtained using the quadratic formula satisfy the original equation, you need to plug the solutions back into the original equation and simplify the expression. If the equation holds true for both solutions, then you can be confident that you have found the correct solutions.

Q: What is the significance of the solutions obtained using the quadratic formula?

A: The solutions obtained using the quadratic formula represent the values of $x$ that satisfy the original equation. In this case, the solutions $x = 1 + \sqrt{47}$ and $x = 1 - \sqrt{47}$ represent the values of $x$ that satisfy the equation $2x^2 + 3x - 7 = x^2 + 5x + 39$.

Q: Can I use other methods to solve the quadratic equation?

A: Yes, you can use other methods to solve the quadratic equation, such as factoring or completing the square. However, the quadratic formula is a powerful tool that can be used to solve quadratic equations of any form.

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:

• Not identifying the values of $a$, $b$, and $c$ correctly
• Not simplifying the expression correctly
• Not verifying that the solutions satisfy the original equation

Conclusion

In this article, we answered some common questions related to the solution of the quadratic equation $2x^2 + 3x - 7 = x^2 + 5x + 39$. We discussed the quadratic formula, how to apply it, and how to verify that the solutions satisfy the original equation. We also highlighted some common mistakes to avoid when using the quadratic formula.