Solve For { X $}$ In The Equation X 2 + 10 X + 25 = 49 X^2 + 10x + 25 = 49 X 2 + 10 X + 25 = 49 .A. { X = -12, 2 $}$ B. { X = 12, 2 $}$ C. { X = -12, -2 $}$ D. { X = 12, -2 $}$

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. In this article, we will focus on solving a specific quadratic equation, $x^2 + 10x + 25 = 49$, to find the value of $x$. We will use various techniques and methods to solve this equation and provide a step-by-step solution.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = 10$, and $c = 25$. The equation is $x^2 + 10x + 25 = 49$, which can be rewritten as $x^2 + 10x - 24 = 0$. This equation represents a parabola that opens upwards, and we need to find the values of $x$ that satisfy this equation.

Rearranging the Equation

To solve the equation, we need to rearrange it in the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. We can do this by subtracting $49$ from both sides of the equation, which gives us $x^2 + 10x - 24 = 0$.

Factoring the Equation

One way to solve a quadratic equation is to factor it. However, in this case, the equation does not factor easily. We can try to factor the equation by finding two numbers whose product is $-24$ and whose sum is $10$. These numbers are $12$ and $-2$, since $12 \times (-2) = -24$ and $12 + (-2) = 10$. Therefore, we can rewrite the equation as $(x + 12)(x - 2) = 0$.

Solving for $x$

Now that we have factored the equation, we can solve for $x$ by setting each factor equal to zero. This gives us two equations: $x + 12 = 0$ and $x - 2 = 0$. Solving for $x$ in each equation, we get $x = -12$ and $x = 2$.

Conclusion

In conclusion, we have solved the quadratic equation $x^2 + 10x + 25 = 49$ to find the values of $x$. We used various techniques, including rearranging the equation, factoring, and solving for $x$. The final answer is that the values of $x$ are $-12$ and $2$.

Final Answer

The final answer is: $\boxed{A}$

Discussion

The solution to the quadratic equation $x^2 + 10x + 25 = 49$ is $x = -12, 2$. This is the correct answer, and it can be verified by plugging the values of $x$ back into the original equation.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Rearrange the equation: Subtract $49$ from both sides of the equation to get $x^2 + 10x - 24 = 0$.
2. Factor the equation: Factor the equation by finding two numbers whose product is $-24$ and whose sum is $10$. These numbers are $12$ and $-2$, so we can rewrite the equation as $(x + 12)(x - 2) = 0$.
3. Solve for $x$: Set each factor equal to zero and solve for $x$. This gives us two equations: $x + 12 = 0$ and $x - 2 = 0$. Solving for $x$ in each equation, we get $x = -12$ and $x = 2$.

Tips and Tricks

Here are some tips and tricks to help you solve quadratic equations:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Factor the equation: Factoring the equation can be a quick and easy way to solve it. Look for two numbers whose product is $c$ and whose sum is $b$.
• Use the method of substitution: If the equation does not factor easily, you can try using the method of substitution. This involves substituting a variable into the equation and solving for that variable.

Conclusion

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. In this article, we have solved a specific quadratic equation, $x^2 + 10x + 25 = 49$, to find the value of $x$. We used various techniques, including rearranging the equation, factoring, and solving for $x$. The final answer is that the values of $x$ are $-12$ and $2$.

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Introduction

In our previous article, we solved the quadratic equation $x^2 + 10x + 25 = 49$ to find the values of $x$. We used various techniques, including rearranging the equation, factoring, and solving for $x$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving quadratic equations.

Q&A

Q: What is the first step in solving a quadratic equation?

A: The first step in solving a quadratic equation is to rearrange it in the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. This involves subtracting the constant term from both sides of the equation.

Q: How do I factor a quadratic equation?

A: Factoring a quadratic equation involves finding two numbers whose product is $c$ and whose sum is $b$. These numbers are called the factors of the equation. Once you have found the factors, you can rewrite the equation as $(x + p)(x + q) = 0$, where $p$ and $q$ are the factors.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can be used to find the values of $x$ that satisfy the equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. This will give you two possible values of $x$, which you can then use to solve the equation.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, which means that it has a squared variable. A linear equation, on the other hand, is a polynomial equation of degree one, which means that it has a single variable.

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

A: No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is specifically designed to solve quadratic equations, and it will not work for linear equations.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$. The discriminant determines the nature of the solutions to the 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 a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. The quadratic formula will give you two complex solutions, which you can then use to solve the equation.

Conclusion

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. In this article, we have provided a Q&A section to help clarify any doubts and provide additional information on solving quadratic equations. We hope that this article has been helpful in understanding the concepts and techniques involved in solving quadratic equations.

Final Answer

The final answer is: $\boxed{A}$

Discussion

The solution to the quadratic equation $x^2 + 10x + 25 = 49$ is $x = -12, 2$. This is the correct answer, and it can be verified by plugging the values of $x$ back into the original equation.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Rearrange the equation: Subtract $49$ from both sides of the equation to get $x^2 + 10x - 24 = 0$.
2. Factor the equation: Factor the equation by finding two numbers whose product is $-24$ and whose sum is $10$. These numbers are $12$ and $-2$, so we can rewrite the equation as $(x + 12)(x - 2) = 0$.
3. Solve for $x$: Set each factor equal to zero and solve for $x$. This gives us two equations: $x + 12 = 0$ and $x - 2 = 0$. Solving for $x$ in each equation, we get $x = -12$ and $x = 2$.

Tips and Tricks

Here are some tips and tricks to help you solve quadratic equations:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Factor the equation: Factoring the equation can be a quick and easy way to solve it. Look for two numbers whose product is $c$ and whose sum is $b$.
• Use the method of substitution: If the equation does not factor easily, you can try using the method of substitution. This involves substituting a variable into the equation and solving for that variable.

Conclusion

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. In this article, we have solved a specific quadratic equation, $x^2 + 10x + 25 = 49$, to find the value of $x$. We used various techniques, including rearranging the equation, factoring, and solving for $x$. The final answer is that the values of $x$ are $-12$ and $2$.