In this article, we’ll delve into solving the specific quadratic equation ${}^{}$ and explore its implications. Quadratic equations are a fundamental part of algebra, representing a parabolic relationship in a variety of real-world scenarios. Let’s start with the solutions.

## The Quadratic Equation – $x_{2}−11x+28=0$

The general form of a quadratic equation is $ax^_{2}+bx+c=0$. In our case, the equation ${}^{}$ can be identified with $$, $b=−11$, and $$.

### Method 1: Factoring

The equation can be factored if we can find two numbers that multiply to 28 (the constant term) and add up to -11 (the coefficient of x). These numbers are -7 and -4. Therefore, the equation can be factored as:

${}^{}$

Setting each factor equal to zero gives us the solutions:

$$ $$

$$

Hence, the solutions are $\mathrm{}$ and $x=4$.

### Method 2: Completing the Square

To solve by completing the square, rearrange the equation and complete the square on the left side:

- Rearrange the equation: ${}^{}\mathrm{}$
- Add the square of half the coefficient of x to both sides: ${\mathrm{x^2}}^{}-\mathrm{11x}+{\left(\frac{11}{2}\right)}^{2}=-28+{\left(\frac{11}{2}\right)}^{2}$ $⇒$ x^2$-11x+30.25=2.25$
- Write the left side as a perfect square: $(x-5.5{)}^{2}=2.25$
- Take the square root of both sides: x$-5.5=\pm \sqrt{2.25}$
- Solve for x: $x=5.5±1.5$

Thus, the solutions are $$ and $$.

### Method 3: Quadratic Formula

The quadratic formula is $\ufffd=\frac{-b\pm \sqrt{{b}^{2}-4\mathrm{ac}}}{2a}$, where $$, $$, and $$ are coefficients from the quadratic equation $\mathrm{ax^}{2}^{}+\mathrm{bx}+c=0$. For our equation:

- $$
- $$
- $c=28$

Plugging these values into the formula:

$x=\frac{-(-11)\pm \sqrt{(-11{)}^{2}-4\cdot 1\cdot 28}}{2\cdot 1}$

$x=\frac{11\pm \sqrt{121-112}}{2}$

$x=\frac{11\pm \sqrt{9}}{2}$

$x=\frac{11\pm 3}{2}$

Therefore, the solutions are:

$x=\frac{11+3}{2}=7$

$x=\frac{11-3}{2}=4$

The solutions for ${}^{}$ are $$ and $$, irrespective of the method used

Also Read:

- Free Redeem Code Generator; Complete Knowledge
- Which Billionaires Own Bitcoin Today?
- Sad Shayari Punjabi

## Graphical Interpretation

The graph of $y=x_{2}−11x+28$ is a parabola opening upwards. The roots of the equation, $x=4$ and $$, represent the points where this parabola intersects the x-axis.

## Conclusion

Solving quadratic equations like $x_{2}−11x+28=0$ can be approached through various methods, each providing a unique perspective on the problem. The roots of this particular equation, 4 and 7, offer insights into the parabolic nature of quadratic relationships in mathematics and their applications.