In this article, we’ll delve into solving the specific quadratic equation x2−11x+28=0 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 – x2−11x+28=0

The general form of a quadratic equation is ax^2+bx+c=0. In our case, the equation x2−11x+28=0 can be identified with $a=1$, $b=-11$, and $c=28$.

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:

x2−11x+28=(x−7)(x−4)=0

Setting each factor equal to zero gives us the solutions:

$x-7=0\Rightarrow x=7$

$x-4=0\Rightarrow x=4$

Hence, the solutions are $x=7$ 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:

1. Rearrange the equation: x2−11x=−28
2. Add the square of half the coefficient of x to both sides: x^2−11x +(112)2=−28+(112)2 $\Rightarrow$  x^2−11x+30.25=2.25
   
3. Write the left side as a perfect square: (x−5.5)2=2.25
   
4. Take the square root of both sides:  x −5.5=±2.25​
5. Solve for x:  $x=5.5\pm 1.5$

Thus, the solutions are $x=5.5+1.5=7$ and $x=5.5-1.5=4$.

Method 3: Quadratic Formula

The quadratic formula is �=−b ±b2−4ac2a​​, where $a$, $b$, and $c$ are coefficients from the quadratic equation ax^2+bx+c=0. For our equation:

• $a=1$
• $b=-11$
• $c=28$

Plugging these values into the formula:

x=−(−11)±(−11)2−4⋅1⋅282⋅1​​

x=11±121−1122​​

x=11±92​​

x=11±32​

Therefore, the solutions are:

x=11+32=7

x=11−32=4

The solutions for x2−11x+28=0 are $x=7$ and $x=4$, irrespective of the method used

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Graphical Interpretation

The graph of y=x2−11x+28 is a parabola opening upwards. The roots of the equation, $x=4$ and $x=7$, represent the points where this parabola intersects the x-axis.

Conclusion

Solving quadratic equations like x2−11x+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.