September 15, 2024

# How to Solve x2-11x+28=0 Equation?

Solving the quadratic equation x2-11x+28=0 can be approached through several mathematical strategies, each providing a unique lens through which to understand the solution. This article delves deeply into three methods: factoring, completing the square, and the quadratic formula, offering a comprehensive guide to tackling similar problems.

### Introduction to Quadratic Equations x2-11x+28=0

A quadratic equation is a second-degree polynomial equation of Ax2+Bx+C=0, where A, B, and C are constants, and A is not equal to 0. The solutions to these equations, known as roots, can be real or complex numbers. The equation x2-11x+28=0 is a classic example, showcasing how different techniques can be applied to find its roots.

### 1. Factoring Method

Factoring is often the first method attempted when solving quadratic equations, as it can provide the quickest solution if the quadratic easily factors into a product of binomials.

#### Steps for Factoring:

1. Identify the Coefficients: For the given equation, the coefficients are A=1 and C=28.
2. Search for Factors: We seek two numbers whose product equals and the sum equals B. For our equation, these numbers must be multiplied by 28 and added to -11.
3. Express in Factored Form: Once the numbers are found, we rewrite the quadratic equation as (x-p)(x-q)=0, where P and q are the roots.

For x2-11x+28=0, the numbers that satisfy the conditions are 7 and 4, leading to the factored form (x-p)(x-q)=0.

### 2. Completing the Square

This method transforms the quadratic equation into a perfect square trinomial, allowing us to solve for x by taking square roots. It’s beneficial when the equation does not factor easily.

#### Steps for Completing the Square:

1. Rearrange the Equation: Ensure the equation is in the form Ax2+Bx=-C.
2. Form a Perfect Square: Add the square of half the coefficient of x to both sides to create a perfect square on one side of the equation.
3. Solve for x: Take the square root of both sides and solve the resulting linear equation.

For x2-11x+28=0, completing the square would involve adding and subtracting

${\left(\frac{-11}{2}\right)}^{2}$

to form a perfect square trinomial, then solve for x.

The quadratic formula is a universal method that can solve any quadratic equation, providing a direct path to the roots without the need for factoring or completing the square.

#### The Formula:

$x=\frac{-B±\sqrt{{B}^{2}-4\mathrm{AC}}}{2A}$

Applying this formula to x2-11x+28=0, we substitute A=1, and C=28 to find the roots.

### Calculation and Results

Using the quadratic formula for our specific equation, we calculated that the roots are x=7 and x=4. This confirms that the equation can be factored into (x-7)(x-4)=0, aligning with the solutions obtained through the factoring method.

### Comparative Analysis and Conclusion

Each of the methods discussed offers a distinct approach to solving quadratic equations:

• Factoring is most efficient when the quadratic easily decomposes into binomials, as seen with x2-11x+28=0.
• Completing the Square provides a systematic way to solve any quadratic equation, which is especially useful when the equation does not readily factor.
• Quadratic Formula is the most versatile and failsafe method, applicable to all quadratic equations, regardless of their complexity.