What is the Solution to 2x^2 - 5x 3 0?

The quadratic equation 2x^2 - 5x 3 0 can be solved using multiple methods such as factorization, completing the square, and the quadratic formula. In this article, we will explore these methods in detail and explain how to find the solutions x 1 and x 3/2.

1. Solving by Factorization

To solve the quadratic equation 2x^2 - 5x 3 0 by factorization, we start by identifying the coefficients:

Given the equation: 2x^2 - 5x 3 0 Factorize the quadratic expression using the coefficients' product and the sum:

For the equation 2x^2 - 5x 3 0, identify the quadratic coefficient (a 2), the linear coefficient (b -5), and the constant term (c 3).

Find the factors of ac (6) that sum to b (-5): the factors are -2 and -3.

Use these factors to rewrite the middle term and factor by grouping:

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

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

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

Solve for x:

x - 1 0 or 2x - 3 0

x 1 or x 3/2

2. Solving by Completing the Square

Start with the given equation: 2x^2 - 5x 3 0. Isolate the x terms:

2x^2 - 5x -3

Divide both sides by the coefficient of x^2 (2):

x^2 - (5/2)x -3/2

Complete the square on the left-hand side by adding the square of half the coefficient of x, (5/4)^2:

x^2 - (5/2)x (5/4)^2 -3/2 (5/4)^2

(x - 5/4)^2 -3/2 25/16

(x - 5/4)^2 1/16

Take the square root of both sides:

x - 5/4 ±1/4

Isolate x:

x 5/4 ± 1/4

x 6/4 or x 4/4

x 3/2 or x 1

3. Solving Using the Quadratic Formula

Recall the quadratic formula: x [-b ± √(b^2 - 4ac)] / (2a) Substitute the coefficients into the formula for the equation 2x^2 - 5x 3 0:

Given: a 2, b -5, c 3

x [-(-5) ± √((-5)^2 - 4 * 2 * 3)] / (2 * 2)

x [5 ± √(25 - 24)] / 4

x [5 ± √1] / 4

x [5 ± 1] / 4

x 6/4 or x 4/4

x 3/2 or x 1

By exploring these methods, we have successfully found the solutions to the quadratic equation 2x^2 - 5x 3 0. Whether you prefer factorization, completing the square, or the quadratic formula, each method provides a systematic approach to solving quadratic equations.