Solve The Equation:$\[ 36x^2 - 4 = 0 \\]Step 1: Simplify The Equation.$\[ 36x^2 = 4 \\]$\[ 4x + 4x \\] (This Step Seems To Be Misplaced And Should Be Ignored.)

Feb 26, 2025 by ADMIN 160 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 the quadratic equation $36x^2 - 4 = 0$ using a step-by-step approach. We will break down the solution into manageable parts, making it easier to understand and follow along.

Step 1: Simplify the Equation

The first step in solving the quadratic equation is to simplify it by isolating the variable $x$ . To do this, we add $4$ to both sides of the equation, resulting in:

${ 36x^2 = 4 }$

This simplification is a crucial step in solving the equation, as it allows us to focus on the quadratic term and its coefficient.

Step 2: Divide Both Sides by the Coefficient

Next, we need to isolate the variable $x$ by dividing both sides of the equation by the coefficient of the quadratic term, which is $36$ . This gives us:

${ x^2 = \frac{4}{36} }$

We can simplify the fraction on the right-hand side by dividing both the numerator and the denominator by their greatest common divisor, which is $4$ . This results in:

${ x^2 = \frac{1}{9} }$

Step 3: Take the Square Root of Both Sides

Now that we have isolated the variable $x$ by dividing both sides of the equation by the coefficient, we can take the square root of both sides to solve for $x$ . This gives us:

${ x = \pm \sqrt{\frac{1}{9}} }$

We can simplify the square root by recognizing that $\sqrt{\frac{1}{9}} = \frac{1}{3}$ .

Step 4: Simplify the Solution

Finally, we can simplify the solution by combining the positive and negative square roots into a single expression:

${ x = \pm \frac{1}{3} }$

This is the final solution to the quadratic equation $36x^2 - 4 = 0$ .

Discussion

In this article, we have walked through the step-by-step process of solving a quadratic equation. We have simplified the equation, isolated the variable, and taken the square root of both sides to arrive at the final solution. This process is a fundamental concept in mathematics, and it is essential to understand and apply it in various contexts.

Conclusion

Solving quadratic equations is a crucial skill for students and professionals alike. By following the step-by-step approach outlined in this article, we can solve quadratic equations with ease and confidence. Whether you are a student or a professional, this article has provided you with a comprehensive guide to solving quadratic equations.

Additional Resources

For further practice and review, we recommend the following resources:

Khan Academy: Quadratic Equations
Mathway: Quadratic Equation Solver
Wolfram Alpha: Quadratic Equation Solver

By practicing and reviewing these resources, you can improve your skills and become more confident in solving quadratic equations.

Frequently Asked Questions

Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two, which means it has a squared variable.

Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you need to simplify it, isolate the variable, and take the square root of both sides.

Q: What is the difference between a quadratic equation and a linear equation? A: A quadratic equation has a squared variable, while a linear equation has a variable raised to the power of one.

Glossary

Quadratic equation: A polynomial equation of degree two, which means it has a squared variable.
Coefficient: A number that is multiplied by a variable in an equation.
Square root: A number that, when multiplied by itself, gives a specified value.
Isolate: To separate a variable from other terms in an equation.

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 address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand and apply this concept.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a squared variable. It is typically written in the form of $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

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

Simplify the equation by combining like terms.
Isolate the variable by dividing both sides of the equation by the coefficient of the quadratic term.
Take the square root of both sides of the equation to solve for the variable.

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

A: A quadratic equation has a squared variable, while a linear equation has a variable raised to the power of one. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation, while the equation $2x + 3 = 0$ is a linear equation.

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: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is written in the form of:

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

where $a$ , $b$ , and $c$ are the constants in the quadratic equation.

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

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

Identify the values of $a$ , $b$ , and $c$ in the quadratic equation.
Plug these values into the quadratic formula.
Simplify the expression under the square root.
Take the square root of both sides of the equation to solve for the variable.

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

A: A quadratic equation is a polynomial equation of degree two, while a polynomial equation can have any degree. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation, while the equation $x^3 + 2x^2 + 3x + 1 = 0$ is a polynomial equation of degree three.

Q: How do I graph a quadratic equation?

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

Identify the values of $a$ , $b$ , and $c$ in the quadratic equation.
Determine the vertex of the parabola by using the formula $x = -\frac{b}{2a}$ .
Plot the vertex on the coordinate plane.
Use the quadratic formula to find the x-intercepts of the parabola.
Plot the x-intercepts on the coordinate plane.
Draw a smooth curve through the vertex and the x-intercepts to graph the parabola.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding and applying the concepts outlined in this article, you can become more confident in solving quadratic equations and tackle more complex mathematical problems.