September 12, 2024

# Solving − x2-11x+28=0 | Quadratic Equation

Quadratic equations are pivotal in algebra, offering insights into various phenomena across disciplines such as physics, astronomy, economics, and even biology. The equation ${}^{}$ is a classic example, embodying the quintessential properties of quadratic forms. Its analysis not only enlightens us about specific solutions but also reveals broader mathematical concepts.

## Structure and Characteristics of Equation x2-11x+28=0

This quadratic equation falls under the standard form , where , , and  are real numbers, and $\mathrm{a \ne }0$. In ${}^{}$x2-11x+28=0:

• , showing the equation is a simple quadratic without a leading coefficient.
• , indicating the linear term’s influence on the curve’s shift and orientation.
• , the constant, affects the vertical translation of the quadratic graph.

## Comprehensive Solving Techniques

1. Factoring Method: Factoring is often the most straightforward approach. It involves breaking down the quadratic into a product of binomials. In this case, finding two numbers that multiply to 28 (the constant term) and add up to -11 (the coefficient of ) leads to the factors  and . Setting each factor equal to zero, we find the solutions: and .

2. Quadratic Formula: The quadratic formula, $x=\frac{-b±\sqrt{{b}^{2}-4\mathrm{ac}}}{2a}$, is a universal solver for quadratic equations. Applying it to our equation, we calculate , leading to the same roots of   and .

3. Completing the Square: This method involves rearranging the equation into a perfect square plus a constant. It’s a bit more complex but offers a deeper understanding of the equation’s geometry. For , this method also confirms the roots found by the other methods.

## Graphical Insights

Plotting  reveals a parabola opening upwards. This orientation is due to the positive coefficient of x${}^{2}$. The vertex of the parabola provides the equation’s minimum value, and the roots (where the parabola crosses the x-axis) at and are key to understanding the equation’s real-world implications.

## Diverse Applications

• Physics: They model trajectories in projectile motion and the behavior of objects under uniform acceleration.
• Economics: Quadratics can represent various economic behaviors, like the relationship between supply, demand, and pricing.
• Engineering and Architecture: They assist in designing structures with specific stress and load requirements, like arches and bridges, using parabolic curves for optimal strength.

## Conclusion

The equation ${}^{}$ is more than just a mathematical problem; it’s a gateway to understanding a range of natural and man-made phenomena. By dissecting this equation, we uncover fundamental principles that apply across numerous fields, demonstrating the universal language of mathematics.

## FAQs

What are the solutions to the equation x2−11x+28=0?

The solutions to the equation x2−11x+28=0 are x=4 and x=7. These solutions can be found using methods such as factoring, the quadratic formula, or completing the square.

How do you factor the equation x2−11x+28=0?

To factor x2−11x+28=0, look for two numbers that multiply to 28 (the constant term) and add up to -11 (the coefficient of x). The numbers -4 and -7 fit these criteria, so the equation factors to (x−4)(x−7)=0.

Can the quadratic formula be used to solve x2−11x+28=0?

Yes, the quadratic formula, ​​x=2ab±b2−4ac​, can be used to solve x2−11x+28=0. By substituting b=−11, and c=28, the formula yields the solutions x=4 and x=7.

What is the graph of the equation x2−11x+28=0 like?

The graph of x2−11x+28=0 is a parabola that opens upwards. This is because the coefficient of x2 (a = 1) is positive. The roots of the equation, x=4 and x=7, are the points where the parabola intersects the x-axis.