Solve For $x$ In The Equation $x^2 + 10x + 12 = 36$.A. $x = -12$ Or $x = \sqrt{G} 2$B. $x = -11$ Or $x = 1$C. $x = -2$ Or $x = 12$D. $x = -1$ Or $x = 11$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $x^2 + 10x + 12 = 36$, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where a, b, and c are constants. In our given equation, $x^2 + 10x + 12 = 36$, we can rewrite it in the standard form as:

$$x^2 + 10x - 24 = 0$$

The Quadratic Formula

One of the most popular methods for solving quadratic equations is the quadratic formula. The quadratic formula states that for an equation in the form $ax^2 + bx + c = 0$, the solutions are given by:

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

In our given equation, $x^2 + 10x - 24 = 0$, we can identify the values of a, b, and c as:

a = 1, b = 10, and c = -24

Applying the Quadratic Formula

Now that we have identified the values of a, b, and c, we can plug them into the quadratic formula to find the solutions. Substituting the values, we get:

$$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(-24)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{-10 \pm \sqrt{100 + 96}}{2}$$

$$x = \frac{-10 \pm \sqrt{196}}{2}$$

$$x = \frac{-10 \pm 14}{2}$$

Therefore, we have two possible solutions:

$$x = \frac{-10 + 14}{2} = 2$$

$$x = \frac{-10 - 14}{2} = -12$$

Checking the Solutions

To verify that these solutions are correct, we can plug them back into the original equation to check if they satisfy the equation. Substituting x = 2, we get:

$$(2)^2 + 10(2) - 24 = 4 + 20 - 24 = 0$$

This confirms that x = 2 is a valid solution. Substituting x = -12, we get:

$$(-12)^2 + 10(-12) - 24 = 144 - 120 - 24 = 0$$

This confirms that x = -12 is also a valid solution.

Conclusion

In this article, we have solved the quadratic equation $x^2 + 10x + 12 = 36$ using the quadratic formula. We have identified the values of a, b, and c, and plugged them into the quadratic formula to find the solutions. We have also verified that the solutions satisfy the original equation. The solutions to the equation are x = 2 and x = -12.

Final Answer

The final answer is:

A. $x = -12$ or $x = \sqrt{G} 2$

However, we must note that the correct answer is x = 2 or x = -12, not x = $\sqrt{G} 2$. The correct answer is not among the options provided, but we can see that the correct solution is x = 2 or x = -12.

Additional Tips and Tricks

• When solving quadratic equations, it's essential to identify the values of a, b, and c correctly.
• The quadratic formula can be used to solve quadratic equations in the form $ax^2 + bx + c = 0$.
• When applying the quadratic formula, make sure to simplify the expression and identify the two possible solutions.
• To verify the solutions, plug them back into the original equation to check if they satisfy the equation.

Common Mistakes to Avoid

• When solving quadratic equations, it's easy to make mistakes when identifying the values of a, b, and c.
• Make sure to simplify the expression correctly when applying the quadratic formula.
• Don't forget to verify the solutions by plugging them back into the original equation.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

Frequently Asked Questions

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 (in this case, x) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve for x by setting each factor equal to zero.
• Quadratic formula: The quadratic formula states that for an equation in the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations of the form $ax^2 + bx + c = 0$. It states that the solutions are given by:

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

Q: How do I apply the quadratic formula?

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

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. Identify the values of a, b, and c in the quadratic equation.
2. Plug these values into the quadratic formula.
3. Simplify the expression to find the two possible solutions.
4. Verify the solutions by plugging them back into the original equation.

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

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

• Not identifying the values of a, b, and c correctly.
• Not simplifying the expression correctly when applying the quadratic formula.
• Not verifying the solutions by plugging them back into the original equation.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Q: How do I choose between factoring and the quadratic formula?

A: You should choose between factoring and the quadratic formula based on the complexity of the quadratic expression. If the quadratic expression can be factored easily, factoring may be a better option. If the quadratic expression is complex, the quadratic formula may be a better option.

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

A: Yes, the quadratic formula can be used to solve quadratic equations with complex solutions. However, you need to be careful when simplifying the expression to find the two possible solutions.

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

A: To verify the solutions to a quadratic equation, you need to plug the solutions back into the original equation to check if they satisfy the equation.

Q: What are some tips for solving quadratic equations?

A: Some tips for solving quadratic equations include:

• Make sure to identify the values of a, b, and c correctly.
• Simplify the expression correctly when applying the quadratic formula.
• Verify the solutions by plugging them back into the original equation.
• Use the quadratic formula when the quadratic expression is complex.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By understanding the quadratic formula and how to apply it, you can solve quadratic equations with confidence and accuracy. Remember to identify the values of a, b, and c correctly, simplify the expression correctly, and verify the solutions by plugging them back into the original equation.