Solving the Quadratic Equation: The Value of x in x2 - 5x 6 0

Quadratic equations are some of the most fundamental mathematical expressions encountered in algebra. These equations feature a variable x raised to the second power, along with a linear term and a constant. In this article, we explore the process of solving the specific quadratic equation:

Understanding the Problem

Given the quadratic equation x2 - 5x 6 0, our task is to find the values of x that satisfy the equation. This involves finding the roots of the equation and understanding the techniques used to solve it.

Factoring the Polynomial

To solve the equation, we first factorize the quadratic polynomial.

Step 1: Identifying the Coefficients

From the equation x2 - 5x 6 0, we identify the coefficients as:

a 1 (coefficient of x2) b -5 (coefficient of x) c 6 (constant term)

The goal is to find two numbers that multiply to the constant term (c) and add up to the coefficient of x (b). In this case, those numbers are -2 and -3, as -2 * -3 6 and -2 - 3 -5.

Step 2: Factoring the Equation

Using these numbers, we can factor the equation as follows:

x2 - 5x 6 (x - 2)(x - 3) 0

Setting each factor equal to zero gives us:

(x - 2 0 Rightarrow x 2) (x - 3 0 Rightarrow x 3)

Verification

Let's verify if these solutions satisfy the original equation by substituting x 2 and x 3 into the equation:

Verification with (x 2)

[2^2 - 5(2) 6 4 - 10 6 0]

Verification with (x 3)

[3^2 - 5(3) 6 9 - 15 6 0]

Both values satisfy the equation, confirming that the solutions are correct.

Using the Quadratic Formula

For completeness, let’s use the quadratic formula to solve the equation:

Step 1: Applying the Quadratic Formula

The quadratic formula is given by:

[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Substituting the coefficients a 1, b -5, and c 6, we get:

[x frac{5 pm sqrt{(-5)^2 - 4(1)(6)}}{2(1)} frac{5 pm sqrt{25 - 24}}{2} frac{5 pm 1}{2}]

Step 2: Simplifying the Expression

This results in:

[x frac{5 1}{2} frac{6}{2} 3] [x frac{5 - 1}{2} frac{4}{2} 2]

The solutions are again x 2 and x 3.

Concluding Remarks

In summary, the solutions to the quadratic equation x2 - 5x 6 0 are x 2 and x 3. We used both factoring and the quadratic formula to reach this conclusion.

Keywords: quadratic equation, solving quadratic equations, quadratic formula