t^2 - 6t + 8 = 0 - Richter Guitar
Solving the Quadratic Equation: t² – 6t + 8 = 0 — A Complete Guide
Solving the Quadratic Equation: t² – 6t + 8 = 0 — A Complete Guide
Understanding how to solve quadratic equations is a fundamental skill in algebra, essential for students, mathematicians, and professionals in various STEM fields. One classic example is the equation t² – 6t + 8 = 0, a beautifully simple quadratic with real-world applications. In this SEO-optimized article, we will explore how to solve this equation step-by-step, understand its meaning, and discover its practical uses.
Understanding the Context
What is t² – 6t + 8 = 0?
The equation t² – 6t + 8 = 0 is a quadratic equation in standard form:
at² + bt + c = 0, where a = 1, b = –6, and c = 8.
Quadratic equations describe parabolic relationships and are crucial in physics, engineering, economics, and many other disciplines. The solutions (roots) of this equation tell us the values of t where the quadratic function f(t) = t² – 6t + 8 equals zero — i.e., the points where the parabola intersects the t-axis.
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Key Insights
Step-by-Step Solution to t² – 6t + 8 = 0
Method 1: Factoring
Factoring is often the fastest way when the quadratic expression can be broken down into simpler binomials.
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Identify two numbers that multiply to c = 8 and add to b = –6.
These numbers are –2 and –4, since:
(-2) × (-4) = 8
(-2) + (–4) = –6 -
Write the factored form:
t² – 6t + 8 = (t – 2)(t – 4) -
Apply the Zero Product Property:
If (t – 2)(t – 4) = 0, then either:
t – 2 = 0 → t = 2
or t – 4 = 0 → t = 4
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✅ Solutions:
t = 2 and t = 4
Method 2: The Quadratic Formula
For any quadratic at² + bt + c = 0, the solutions are:
t = [–b ± √(b² – 4ac)] / (2a)
Plug in a = 1, b = –6, c = 8:
-
Compute discriminant:
Δ = b² – 4ac = (–6)² – 4(1)(8) = 36 – 32 = 4 -
Take square root of Δ:
√Δ = √4 = 2
-
Plug into formula:
t = [–(–6) ± 2] / (2×1) = [6 ± 2] / 2 -
Calculate both solutions:
t = (6 + 2)/2 = 8/2 = 4
t = (6 – 2)/2 = 4/2 = 2
Result: t = 2 and t = 4 — same as before.
