Optimizing Solutions with Quadratic Equations: The Case of \( a = 1 \), \( b = -5 \), \( c = 6 \)

Solving quadratic equations is a fundamental skill in algebra, with applications across science, engineering, economics, and beyond. In this article, we explore a specific quadratic equation with the coefficients \( a = 1 \), \( b = -5 \), and \( c = 6 \), demonstrating step-by-step how to find its roots and understand its significance.

Understanding the Context

The Quadratic Equation: A Closer Look

Start with the standard form of a quadratic equation:

\[
ax^2 + bx + c = 0
\]

Given:
\( a = 1 \),
\( b = -5 \),
\( c = 6 \)

Solved 1. In ABC we have A1∈BC,B1∈(AC),C1∈(AB). If | Chegg.com

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Solved 1. In ABC we have A1∈BC,B1∈(AC),C1∈(AB). If | Chegg.com
Solved Q6/F1=A'B'+B'C+ABC+A'C' F2=A′+B′C+A′C F3 =B′+ABC F4 | Chegg.com
Solved a₁ b₁ C1 b₂ C2 C₂ = 6. Find each of the following. | Chegg.com
Solved (b) (2) (6 marks) Let A=(a1,a2),B=(b1,b2) and | Chegg.com
View question - (a+b+c)²=a²+b²+c² Proved that 1/a+1/b+1/c=0

Key Insights

Substituting these values, the equation becomes:

\[
x^2 - 5x + 6 = 0
\]

This quadratic equation opens upward because the coefficient of \( x^2 \) is positive (\( a = 1 > 0 \)), meaning it has a minimum point at its vertex.

Solving the Equation: Step-by-Step

Continue Reading

Perhaps the 70% is of total visits, and we sum the successful ones, but without location, impossible.
Therefore, in math competition context, accept the expected value as 1146.6, but since answer must be integer, and others are exact, perhaps the problem intends: compute the total number as 1147 (rounding up).
But better: use the exact calculation and box 1146.6? But format is integer.

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Final Thoughts

Method 1: Factoring

Since \( a = 1 \), the equation simplifies nicely to a factorable form. We seek two numbers that multiply to \( c = 6 \) and add to \( b = -5 \).

  • The pair: \( -2 \) and \( -3 \)
    Because:
    \( (-2) \ imes (-3) = 6 \)
    \( (-2) + (-3) = -5 \)

Thus, we factor:

\[
x^2 - 5x + 6 = (x - 2)(x - 3) = 0
\]

Set each factor equal to zero:

\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]

Method 2: Quadratic Formula

For broader understanding, the quadratic formula expresses solutions directly:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]