The Quadratic Equation
Solving \(ax^2 + bx + c = 0\) by completing the square, the discriminant and the quadratic formula, Vieta's relations, and symmetric systems.
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Learning Objectives
- Solve \(ax^2 + bx + c = 0\) by completing the square and with the quadratic formula.
- Count the real roots of a quadratic equation from the sign of the discriminant \(\Delta = b^2 - 4ac\).
- State and apply Vieta's relations \(x_1 + x_2 = -\tfrac{b}{a},\; x_1 x_2 = \tfrac{c}{a}\).
- Form a quadratic equation with prescribed roots, and determine the sign of the roots without solving.
- Solve symmetric systems by the substitution \(s = x + y,\; p = xy\) and the construction of a quadratic equation.
About This Course Unit
We solve the quadratic equation by completing the square, obtaining the discriminant and the quadratic formula. Vieta's relations connect the roots to the coefficients, letting us form equations from known roots, study the sign of the roots without solving, and solve symmetric systems.