16x^2 - 40x + 25
Understanding the Quadratic Equation: 16x² - 40x + 25
Solving quadratic equations is a fundamental skill in algebra, and one expression that exemplifies key concepts in quadratic analysis is 16x² - 40x + 25. Whether you're a student, educator, or math enthusiast, understanding this equation’s structure, solutions, and applications can enhance your problem-solving abilities. In this SEO-optimized article, we’ll explore everything you need to know about 16x² - 40x + 25, including factoring, the quadratic formula, vertex form, and real-world applications.
What is the Equation 16x² - 40x + 25?
The expression 16x² - 40x + 25 is a quadratic trinomial in the standard form:
ax² + bx + c, where a = 16, b = -40, and c = 25.
Quadratic equations of this form appear frequently in algebra, physics, engineering, and economics, making familiarity with them essential. The discriminant, b² - 4ac, helps determine the nature of the roots—whether real and distinct, real and repeated, or complex.
Step 1: Determine the Type of Quadratic Using the Discriminant
Calculate the discriminant:
> D = b² - 4ac = (-40)² - 4(16)(25) = 1600 - 1600 = 0
Since D = 0, the equation has exactly one real root (a repeated root), meaning the parabola touches the x-axis at its vertex.
Factoring the Quadratic Expression
Because the discriminant is zero, a perfect square trinomial is likely. Let's check if 16x² - 40x + 25 factors nicely:
Try factoring:
We seek two binomials of the form (mx + n)² = m²x² + 2mnx + n²
From the equation:
- First term: 16x² → m² = 16 ⇒ m = 4
- Last term: 25 = 5²
- Middle term: 2mnx = -40x
Try m = 4, n = -5:
2mn = 2(4)(-5) = -40 ⇒ matches!
So,
16x² - 40x + 25 = (4x - 5)²
This perfect square form reveals the vertex and simplifies graphing.
Solving for x: Finding the Root
Set the factored expression equal to zero:
(4x - 5)² = 0
Take the square root of both sides:
4x - 5 = 0
Solve:
4x = 5 → x = 5/4
So the equation has a double root at x = 5/4 (or 1.25). This is the x-coordinate of the vertex and the point where the parabola touches the x-axis.
Vertex Form
From (4x - 5)² = 0, expand to get vertex form:
y = (4x - 5)² → vertex at (5/4, 0)
This confirms the equation has vertex (5/4, 0) and opens upward (since the coefficient of x² is positive).
Applications and Real-World Context
The equation 16x² - 40x + 25 models situations where outcomes peak at a single point, such as:
- Projectile motion: Maximum height achieved when vertical position follows a quadratic (though usually without damping).
- Optimization: Cost or revenue functions with a single maximum or minimum.
- Geometry: Areas or distances expressed via quadratic relationships.
For example, if modeling the area of a rectangle with adjustable sides, this form might emerge from algebraic manipulation.
Using the Quadratic Formula
For reinforcement, use the quadratic formula:
> x = [-b ± √(b² - 4ac)] / (2a)
Plug in a = 16, b = -40, c = 25:
x = [40 ± √((-40)² - 4(16)(25))] / (32) = [40 ± √(1600 - 1600)] / 32 = [40 ± 0] / 32 = 40 / 32 = 5/4
This matches our earlier result—consistent and accurate.
Conclusion
The quadratic expression 16x² - 40x + 25 is a perfect square trinomial with a repeated real root at x = 5/4. Its factorization as (4x - 5)² reveals both the solution and the parabola’s vertex. Understanding this equation strengthens foundational algebra skills while offering insights into modeling real-world scenarios with quadratics.
Whether you’re factoring, solving with the quadratic formula, or analyzing the vertex form, mastering such equations is crucial for advanced math and STEM fields.
Summary Table: | Feature | Value/Result | |---------------------------|-------------------------------| | Form | Perfect square trinomial | | Factored Form | (4x - 5)² | | Root(s) | x = 5/4 (double root) | | Vertex (x, y) | (5/4, 0) | | Axis of Symmetry | x = 5/4 | | Discriminant | D = 0 (repeated root) | | Best Used For | Perfect square scenarios, vertex form, optimization |
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Start mastering quadratics today—perfect squares open the door to elegant solutions!
