Solve For \[$ X \$\] In The Equation \[$ H(x) = 4 \$\].Given: \[$ H(x) = X^2 - 5 \$\].

Introduction

In mathematics, solving quadratic equations is a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on solving quadratic equations of the form ${h(x) = x^2 - 5}$, where we need to find the value of ${x}$ that satisfies the equation ${h(x) = 4}$.

Understanding the Equation

The given equation is ${h(x) = x^2 - 5}$, and we are asked to solve for ${x}$ when ${h(x) = 4}$. This means we need to find the value of ${x}$ that makes the equation ${x^2 - 5 = 4}$ true. To do this, we can start by adding 5 to both sides of the equation, which gives us ${x^2 = 9}$.

Solving for ${x}$

Now that we have the equation ${x^2 = 9}$, we can solve for ${x}$ by taking the square root of both sides. This gives us ${x = \pm \sqrt{9}}$. Since ${\sqrt{9} = 3}$, we have two possible solutions: ${x = 3}$ and ${x = -3}$.

Verifying the Solutions

To verify our solutions, we can plug them back into the original equation ${h(x) = x^2 - 5}$. If we substitute ${x = 3}$, we get ${h(3) = 3^2 - 5 = 9 - 5 = 4}$, which is true. Similarly, if we substitute ${x = -3}$, we get ${h(-3) = (-3)^2 - 5 = 9 - 5 = 4}$, which is also true.

Conclusion

In this article, we have solved the quadratic equation ${h(x) = x^2 - 5}$ for ${x}$ when ${h(x) = 4}$. We started by adding 5 to both sides of the equation, which gave us ${x^2 = 9}$. Then, we took the square root of both sides to get ${x = \pm \sqrt{9}}$. Finally, we verified our solutions by plugging them back into the original equation.

Tips and Tricks

• When solving quadratic equations, it's essential to check your solutions by plugging them back into the original equation.
• If you're given a quadratic equation in the form ${ax^2 + bx + c = 0}$, you can use the quadratic formula to solve for ${x}$: ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$.
• Quadratic equations can be used to model real-world problems, such as the motion of objects under the influence of gravity or the growth of populations.

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, such as the trajectory of a projectile or the motion of a pendulum.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as the supply and demand of goods and services.

Common Mistakes

• Not checking solutions: It's essential to check your solutions by plugging them back into the original equation to ensure they are correct.
• Not using the correct formula: Make sure to use the correct formula for solving quadratic equations, such as the quadratic formula.
• Not considering all possible solutions: Quadratic equations can have multiple solutions, so make sure to consider all possible solutions.

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 is two. It is typically written in the form ${ax^2 + bx + c = 0}$, where ${a}$, ${b}$, and ${c}$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and graphing. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is written as: ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$, where ${a}$, ${b}$, and ${c}$ are the coefficients of the quadratic 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. Then, simplify the expression and solve for ${x}$.

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 ${a}$, ${b}$, and ${c}$ into the quadratic formula.
2. Simplify the expression.
3. Solve for ${x}$.
4. Check your 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 checking solutions
• Not using the correct formula
• Not considering all possible solutions
• Not simplifying the expression correctly

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original equation and simplify. If the equation is true, then your solution is correct.

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, such as the trajectory of a projectile or the motion of a pendulum.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as the supply and demand of goods and services.

Q: Can I use a calculator to solve quadratic equations?

A: Yes, you can use a calculator to solve quadratic equations. However, it's essential to understand the underlying math and be able to check your solutions.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or a computer program. You can also use a table of values to plot the graph.

Q: What are some common types of quadratic equations?

A: Some common types of quadratic equations include:

• Linear quadratic equations: ${ax^2 + bx + c = 0}$
• Quadratic equations with complex roots: ${ax^2 + bx + c = 0}$
• Quadratic equations with rational roots: ${ax^2 + bx + c = 0}$

Q: Can I use quadratic equations to model real-world problems?

A: Yes, you can use quadratic equations to model real-world problems. Quadratic equations can be used to model the behavior of physical systems, such as the motion of objects under the influence of gravity, or the growth of populations.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics that have numerous real-world applications. By understanding the basics of quadratic equations and how to solve them, you can apply them to a wide range of problems. Remember to check your solutions, use the correct formula, and consider all possible solutions.