Solving The Quadratic Equation 4x^2 - 8x + 3 = 0

Jul 13, 2025 by Admin 49 views

In this comprehensive guide, we delve into the methods for finding solutions to the quadratic equation $4x^2 - 8x + 3 = 0$ . Quadratic equations, characterized by the general form $ax^2 + bx + c = 0$ , are fundamental in mathematics and have widespread applications in various fields, including physics, engineering, and economics. We will explore various techniques to solve this equation, including factoring, the quadratic formula, and completing the square, and verify the solutions provided.

Understanding Quadratic Equations

Before we dive into solving the equation, it's crucial to understand the properties of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, x) is 2. The coefficients a, b, and c are constants, and the solutions to the equation are the values of x that make the equation true. These solutions are also known as the roots or zeros of the quadratic equation.

The discriminant, given by the formula $b^2 - 4ac$ , plays a vital role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it's zero, the equation has one real root (a repeated root). If it's negative, the equation has two complex roots. In our case, a = 4, b = -8, and c = 3, so the discriminant is $(-8)^2 - 4 * 4 * 3 = 64 - 48 = 16$ . Since the discriminant is positive, we expect two distinct real roots.

Method 1: Factoring the Quadratic Equation

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be easily factored. The goal is to rewrite the quadratic expression as a product of two binomials. To factor $4x^2 - 8x + 3 = 0$ , we look for two numbers that multiply to (4 * 3 = 12) and add up to -8. These numbers are -6 and -2. We can then rewrite the middle term, -8x, as -6x - 2x:

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

Now, we factor by grouping:

$2x(2x - 3) - 1(2x - 3) = 0$

Notice that $(2x - 3)$ is a common factor. Factoring it out, we get:

$(2x - 1)(2x - 3) = 0$

For the product of two factors to be zero, at least one of them must be zero. Thus, we set each factor equal to zero and solve for x:

$2x - 1 = 0$ or $2x - 3 = 0$

Solving for x in each case, we get:

$2x = 1$ => $x = rac{1}{2}$

$2x = 3$ => $x = rac{3}{2}$

Therefore, the solutions obtained by factoring are $x = rac{1}{2}$ and $x = rac{3}{2}$ .

Method 2: Using the Quadratic Formula

The quadratic formula is a universally applicable method for solving quadratic equations, regardless of whether they can be factored easily. It provides a direct solution for x in terms of the coefficients a, b, and c:

$x = rac{-b

Applying the quadratic formula to our equation, $4x^2 - 8x + 3 = 0$ , where a = 4, b = -8, and c = 3, we have:

$x = rac{-(-8)

Simplifying the expression, we get:

$x = rac{8

Calculating the square root:

$x = rac{8

Now we have two possible solutions:

$x_1 = rac{8 + 4}{8} = rac{12}{8} = rac{3}{2}$

$x_2 = rac{8 - 4}{8} = rac{4}{8} = rac{1}{2}$

As we can see, the quadratic formula yields the same solutions as factoring: $x = rac{1}{2}$ and $x = rac{3}{2}$ .

Method 3: Completing the Square

Completing the square is another powerful method for solving quadratic equations. It involves manipulating the equation to form a perfect square trinomial on one side. While it's not always the most efficient method for simple equations, it's a valuable technique for understanding the structure of quadratic equations and deriving the quadratic formula.

To solve $4x^2 - 8x + 3 = 0$ by completing the square, we first divide the entire equation by the leading coefficient, 4:

$x^2 - 2x + rac{3}{4} = 0$

Next, we move the constant term to the right side of the equation:

$x^2 - 2x = - rac{3}{4}$

To complete the square, we take half of the coefficient of the x term (-2), square it ((-1)^2 = 1), and add it to both sides of the equation:

$x^2 - 2x + 1 = - rac{3}{4} + 1$

Now, the left side is a perfect square trinomial:

$(x - 1)^2 = rac{1}{4}$

Taking the square root of both sides:

$x - 1 =

Solving for x, we get:

$x = 1

This gives us two solutions:

$x_1 = 1 + rac{1}{2} = rac{3}{2}$

$x_2 = 1 - rac{1}{2} = rac{1}{2}$

Again, we obtain the same solutions: $x = rac{1}{2}$ and $x = rac{3}{2}$ .

Verifying the Solutions

To ensure the accuracy of our solutions, it's always a good practice to substitute them back into the original equation. Let's verify $x = rac{1}{2}$ and $x = rac{3}{2}$ :

For $x = rac{1}{2}$ :

$4( rac{1}{2})^2 - 8( rac{1}{2}) + 3 = 4( rac{1}{4}) - 4 + 3 = 1 - 4 + 3 = 0$

For $x = rac{3}{2}$ :

$4( rac{3}{2})^2 - 8( rac{3}{2}) + 3 = 4( rac{9}{4}) - 12 + 3 = 9 - 12 + 3 = 0$

Both solutions satisfy the original equation, confirming that they are correct.

Conclusion: Solutions to the Equation

In conclusion, by employing factoring, the quadratic formula, and completing the square, we have determined the solutions to the quadratic equation $4x^2 - 8x + 3 = 0$ . The solutions are $x = rac{1}{2}$ and $x = rac{3}{2}$ . These solutions were also verified by substituting them back into the original equation.

Therefore, the correct options from the given choices are:

A. $x = rac{1}{2}$
D. $x = rac{3}{2}$

Understanding these methods for solving quadratic equations is crucial for success in algebra and related fields. Mastering these techniques allows for solving a wide range of mathematical problems and real-world applications. Remember to practice and apply these methods to various quadratic equations to solidify your understanding.

Additional Tips for Solving Quadratic Equations

Always check the discriminant first to understand the nature of the roots.
Look for factoring opportunities as it's often the quickest method.
The quadratic formula is a reliable method for any quadratic equation.
Completing the square provides a deeper understanding of quadratic equation structure.
Verify your solutions by substituting them back into the original equation.

By following these tips and practicing regularly, you can confidently solve quadratic equations and excel in your mathematical endeavors. Keep exploring, keep learning, and embrace the power of mathematics! Understanding quadratic equations is a building block for more advanced concepts, so mastering these skills will benefit you in the long run. Don't hesitate to seek help if you encounter difficulties, and remember that practice makes perfect. Happy solving!