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Problem StatementSymbolically Verified

Solve the quadratic equation: 2x^2 - 7x + 3 = 0

Approach:We will solve by factoring the quadratic trinomial into two linear binomials (2x - 1)(x - 3) = 0 and applying the Zero Product Property.

Step-by-Step Rigorous Derivation (3 Steps)

Step Verified

For the general quadratic form $a x^{2} + b x + c = 0$ , we identify: $a = 2$ , $b = - 7$ , and $c = 3$ .

a \cdot c = 2 \times 3 = 6, \quad b = -7

Justification:

Find two integers that multiply to $a c = 6$ and sum to $b = - 7$ . These numbers are $- 6$ and $- 1$ .

Step Verified

Decompose $- 7 x$ as $- 6 x - x$ and group terms pair-wise:

2 x^{2} - 6 x - x + 3 = 2 x (x - 3) - 1 (x - 3) = (2 x - 1) (x - 3)

Justification:

Extract the common binomial factor $(x - 3)$ from both grouped pairs.

Step Verified

Set each independent linear factor equal to zero and isolate $x$ :

2x - 1 = 0 \implies x = \frac{1}{2}, \quad x - 3 = 0 \implies x = 3

Justification:

The two distinct real roots of the quadratic equation are $x = \frac{1}{2}$ and $x = 3$ .

Final Exact Solution

x = \frac{1}{2}, \quad x = 3

Curve & Geometric VerificationFormula: 2*x^2 - 7*x + 3

Canvas: f(x) = 2*x^2 - 7*x + 3

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