Understanding the Quadratic Equation: x² – (a + b)x + ab = 0 and Its Real-World Root: x² – 4x + 3 = 0

When studying quadratic equations, few examples illustrate both algebraic structure and elegant solutions like x² – (a + b)x + ab = 0. This general form reveals hidden patterns that simplify to specific equations—such as x² – 4x + 3 = 0—whose roots offer powerful insights into factoring, solution methods, and applications.

Breaking Down the General Quadratic Form

Understanding the Context

The quadratic equation
x² – (a + b)x + ab = 0
is a carefully constructed identity. It forms a perfect factorable trinomial representing the product of two binomials:
(x – a)(x – b) = 0

Expanding this gives:
x² – (a + b)x + ab
confirming the equivalence.

This structure allows for easy root identification—x = a and x = b—without requiring the quadratic formula, making it a cornerstone in algebraic problem-solving.

How x² – 4x + 3 = 0 Emerges from the General Form

Solved: a) x^2-16=0 (b) x^2+2 x=0 c) x^2-5 x+6=0 [algebra]

Image Gallery

Solved: a) x^2-16=0 (b) x^2+2 x=0 c) x^2-5 x+6=0 [algebra]
Solved: x^2+4x-5=0 b) 2x^2-9x+4=0 c) x^2+8x+16=0 [Math]
a) $(x+2)(x+3) = 0$ b) $(x-1)(x-2) = 40$ | StudyX
Solved Solve for x : (a) x2−2x−3=0 (b) −x2+3x+12=2 (c) | Chegg.com
Solved: a^2+2ab+b^2-x^2 [Math]

Key Insights

Observe that x² – 4x + 3 = 0 matches the general form when:

  • a + b = 4
  • ab = 3

These conditions lead to a powerful deduction: the values of a and b must be the roots of the equation and therefore real numbers satisfying these constraints.

To find a and b, solve for two numbers whose sum is 4 and product is 3.

Step-by-step root calculation:

We solve:
a + b = 4
ab = 3

Continue Reading

The altitude corresponding to a side is $ h = \frac{2A}{\text{side}} $. The longest altitude corresponds to the shortest side, which is 13 cm:
h_{\text{longest}} = \frac{2 \cdot 84}{13} = \frac{168}{13} \approx 12.92
Thus, the length of the longest altitude is $ \boxed{\frac{168}{13}} $ cm.

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Final Thoughts

Using the quadratic property:
If a and b are roots, they satisfy:
t² – (a + b)t + ab = 0 → t² – 4t + 3 = 0

This matches the given equation. Factoring:
(t – 1)(t – 3) = 0

So, the roots are t = 1 and t = 3 → a = 1, b = 3 (or vice versa).

Thus, x² – 4x + 3 = 0 becomes the specific equation with known, easily verifiable roots.

Solving x² – 4x + 3 = 0

Apply factoring:
x² – 4x + 3 = (x – 1)(x – 3) = 0

Set each factor to zero:
x – 1 = 0 → x = 1
x – 3 = 0 → x = 3

Roots are x = 1 and x = 3, reinforcing how x² – (a + b)x + ab = 0 generalizes to concrete solutions when coefficients satisfy real, distinct solutions.

Why This Equation Matters: Applications and Insights

Understanding how general forms reduce to specific equations helps in: