Solving and Standardizing the Quadratic Equation 2x-1^2 3x^2

When it comes to solving quadratic equations, understanding the standard form is crucial. In this article, we will explore the process of solving the quadratic equation 2x-1^2 3x^2 and converting it to the standard form. Let's dive in and understand each step involved in the solution.

Introduction to Quadratic Equations

Quadratic equations are polynomial equations of the second degree. They are generally written in the form ax^2 bx c 0, where a, b, and c are constants, and a ≠ 0. The standard form of a quadratic equation is essential for solving it and finding its roots.

Solving the Given Quadratic Equation

Let's start with the equation 2x-1^2 3x^2.

Step 1: Expand and Simplify

First, we need to expand and simplify the equation. The given equation is:

2x-1^2 3x^2

Expanding the left-hand side, we have:

2x^2 - 1^2 3x^2

Step 2: Rearrange the Equation

Next, we need to rearrange the equation to the standard form ax^2 bx c 0. Let's move all terms to one side of the equation:

2x^2 - 1^2 - 3x^2 0

Simplify by combining like terms:

2x^2 - 3x^2 - 1^2 0

Which simplifies to:

-x^2 - 2 0

Step 3: Multiply Both Sides by -1

To convert the equation to the standard form, we multiply both sides by -1:

x^2 2 0

This can be rewritten as:

x^2 2x 1 - 1 0

Simplify further:

x^2 2x 1 - 1 0

Which leads to:

x^2 2x 1 - 1 0

Simplifying again:

x^2 2x - 5 0

Converting to Standard Form

To convert the given equation to standard form ax^2 bx c 0, we need to identify the coefficients a, b, and c.

Step 1: Identify Coefficients

From the given equation, we can identify the coefficients:

a 1

b 2

c -1

Thus, the standard form of the equation is:

x^2 2x - 1 0

Solving for x

Now that we have the equation in standard form, we can solve for x using the quadratic formula:

Step 1: Use the Quadratic Formula

The quadratic formula is given by:

x [-b ± sqrt(b^2 - 4ac)] / (2a)

For our equation, the coefficients are:

a 1

b 2

c -1

Step 2: Substitute the Values

Substitute the values into the quadratic formula:

x [-2 ± sqrt((2)^2 - 4(1)(-1))] / (2(1))

Simplify the expression:

x [-2 ± sqrt(4 4)] / 2

x [-2 ± sqrt(8)] / 2

x [-2 ± 2sqrt(2)] / 2

Simplify further:

x -1 ± sqrt(2)

Thus, the solutions are:

x -1 sqrt(2)

x -1 - sqrt(2)

Conclusion

In this article, we explored the process of solving and standardizing the quadratic equation 2x-1^2 3x^2. We discussed how to expand and simplify the equation, rearrange it to the standard form, and solve for x using the quadratic formula. By understanding these steps, you can solve similar quadratic equations and convert them to the standard form for easier solving.