Solve For \[$x\$\].$\[4x^2 = 9\\]

Feb 26, 2025 by ADMIN 34 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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 quadratic equations of the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. We will use the given equation $4x^2 = 9$ as an example to demonstrate the steps involved in solving quadratic equations.

Understanding Quadratic Equations

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 $a$ , $b$ , and $c$ are constants. The variable $x$ is the unknown value that we want to solve for.

The Given Equation

The given equation is $4x^2 = 9$ . To solve for $x$ , we need to isolate the variable $x$ on one side of the equation. We can start by subtracting $9$ from both sides of the equation to get $4x^2 - 9 = 0$ .

Step 1: Move the Constant Term to the Right Side

We can rewrite the equation as $4x^2 = 9$ . To move the constant term to the right side, we subtract $9$ from both sides of the equation.

# Given equation
equation = "4x^2 = 9"

# Subtract 9 from both sides
new_equation = equation.replace("9", "-9")
print(new_equation)

Step 2: Factor the Quadratic Expression

The quadratic expression $4x^2$ can be factored as $4(x^2)$ . We can rewrite the equation as $4(x^2) = 9$ .

# Factor the quadratic expression
factored_equation = "4(x^2) = 9"
print(factored_equation)

Step 3: Divide Both Sides by 4

To isolate the variable $x^2$ , we can divide both sides of the equation by $4$ . This gives us $x^2 = \frac{9}{4}$ .

# Divide both sides by 4
simplified_equation = "x^2 = 9/4"
print(simplified_equation)

Step 4: Take the Square Root of Both Sides

To solve for $x$ , we can take the square root of both sides of the equation. This gives us $x = \pm \sqrt{\frac{9}{4}}$ .

# Take the square root of both sides
solution = "x = ±√(9/4)"
print(solution)

Step 5: Simplify the Solution

We can simplify the solution by evaluating the square root of $\frac{9}{4}$ . This gives us $x = \pm \frac{3}{2}$ .

# Simplify the solution
final_solution = "x = ±3/2"
print(final_solution)

Conclusion

Solving quadratic equations involves several steps, including moving the constant term to the right side, factoring the quadratic expression, dividing both sides by the coefficient of the quadratic term, taking the square root of both sides, and simplifying the solution. By following these steps, we can solve quadratic equations of the form $ax^2 + bx + c = 0$ . In this article, we used the given equation $4x^2 = 9$ as an example to demonstrate the steps involved in solving quadratic equations.

Common Quadratic Equations

Here are some common quadratic equations that you may encounter:

$x^2 + 4x + 4 = 0$
$x^2 - 6x + 8 = 0$
$x^2 + 2x - 3 = 0$

Tips and Tricks

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

Make sure to move the constant term to the right side of the equation.
Factor the quadratic expression if possible.
Divide both sides by the coefficient of the quadratic term.
Take the square root of both sides.
Simplify the solution.

Practice Problems

Here are some practice problems to help you practice solving quadratic equations:

Solve the equation $x^2 + 2x - 3 = 0$ .
Solve the equation $x^2 - 6x + 8 = 0$ .
Solve the equation $x^2 + 4x + 4 = 0$ .

Real-World Applications

Quadratic equations have many 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 bridges and other structures.
Economics: Quadratic equations are used to model the behavior of economic systems.

Conclusion

Frequently Asked Questions

Here are some frequently asked questions about 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 $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to follow these steps:

Move the constant term to the right side of the equation.
Factor the quadratic expression if possible.
Divide both sides by the coefficient of the quadratic term.
Take the square root of both sides.
Simplify the solution.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable, while a linear equation does not.

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

A: Yes, you can use a calculator to solve a quadratic equation. However, it's always a good idea to understand the steps involved in solving a quadratic equation, so you can verify the solution and apply it to real-world problems.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you need to examine the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

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

A: Yes, you can use the quadratic formula to solve a quadratic equation. The quadratic formula is:

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

This formula gives you the two solutions to the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that gives you the solutions to a quadratic equation. The formula is:

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

This formula is derived from the fact that a quadratic equation can be factored as $(x - r)(x - s) = 0$ , where $r$ and $s$ are the roots of the equation.

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 the complex solutions to the equation.

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

A: To apply the quadratic formula to a quadratic equation, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, you need to simplify the expression and solve for $x$ .

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. It determines the number of solutions to the quadratic equation. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated 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 a negative discriminant?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a negative discriminant. However, the solutions will be complex numbers.

Q: How do I determine the nature of the solutions to a quadratic equation?

A: To determine the nature of the solutions to a quadratic equation, you need to examine the discriminant. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one repeated 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 a repeated solution?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a repeated solution. The quadratic formula will give you the repeated solution.

Q: How do I apply the quadratic formula to a quadratic equation with a repeated solution?

A: To apply the quadratic formula to a quadratic equation with a repeated solution, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, you need to simplify the expression and solve for $x$ .

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

A: The quadratic formula is a fundamental concept in mathematics that gives you the solutions to a quadratic equation. It is used in a wide range of applications, including physics, engineering, and economics.

Q: Can I use the quadratic formula to solve a quadratic equation with a complex solution?

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

Q: How do I apply the quadratic formula to a quadratic equation with a complex solution?

A: To apply the quadratic formula to a quadratic equation with a complex solution, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, you need to simplify the expression and solve for $x$ .

Q: What is the significance of the quadratic formula in real-world applications?

A: The quadratic formula is used in a wide range of real-world applications, including physics, engineering, and economics. It is used to model the behavior of complex systems and to solve problems that involve quadratic equations.

Q: Can I use the quadratic formula to solve a quadratic equation with a negative coefficient?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a negative coefficient. The quadratic formula will give you the solutions to the equation.

Q: How do I apply the quadratic formula to a quadratic equation with a negative coefficient?

A: To apply the quadratic formula to a quadratic equation with a negative coefficient, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, you need to simplify the expression and solve for $x$ .

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

A: The quadratic formula is a fundamental concept in algebra that gives you the solutions to a quadratic equation. It is used to solve problems that involve quadratic equations and to model the behavior of complex systems.

Q: Can I use the quadratic formula to solve a quadratic equation with a complex coefficient?

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

Q: How do I apply the quadratic formula to a quadratic equation with a complex coefficient?

A: To apply the quadratic formula to a quadratic equation with a complex coefficient, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, you need to simplify the expression and solve for $x$ .

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

A: The quadratic formula is used in calculus to solve problems that involve quadratic equations and to model the behavior of complex systems.

Q: Can I use the quadratic formula to solve a quadratic equation with a negative discriminant?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a negative discriminant. However, the solutions will be complex numbers.

Q: How do I determine the nature of the solutions to a quadratic equation with a negative discriminant?

A: To determine the nature of the solutions to a quadratic equation with a negative discriminant, you need to examine the discriminant. If the discriminant is negative, the equation has no real solutions.

Q: Can I use the quadratic formula to solve a quadratic equation with a repeated solution and a negative discriminant?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a repeated solution and a negative discriminant. However, the solutions will be complex numbers.

Q: How do I apply the quadratic formula to a quadratic equation with a repeated solution and a negative discriminant?

A: To apply the quadratic formula to a quadratic equation with a repeated solution and a negative discriminant, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, you need to simplify the expression and solve for $x$ .

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

A: The quadratic formula is used in statistics to solve problems that involve quadratic equations and to model the behavior of complex systems.

Q: Can I use the quadratic formula to solve a quadratic equation with a complex coefficient and a negative discriminant?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a complex coefficient and a negative discriminant. However, the solutions will be complex numbers.

Q: How do I determine the nature of the solutions to a quadratic equation with a complex coefficient and a negative discriminant?

A: To determine the nature of the solutions to a quadratic equation with a