Quadratic Equations
Quadratic Equations - JEE & CBSE
Concepts:
- Quadratic equation:
Equation in the form of $$ ax^2 + bx + c = 0$$, where $$a \neq 0$$
- Standard form:
Rearranged form of the equation as $$ax^2 + bx + c = 0$$
- Roots of quadratic equations:
Values of $$x$$ that make $$ax^2 + bx + c = 0$$ true
- Discriminant (D):
Quantity under the square root in the quadratic formula, $$D = b^2 - 4ac$$
- Determines nature of roots:
- $$D > 0$$, real and distinct roots
- $$D = 0$$, equal roots
- $$D < 0$$, no real roots (complex and conjugate)
- Quadratic formula: $$ x = \frac{-b \pm \sqrt{D}}{2a}$$
Results:
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Sum of roots: $$ x_1 + x_2 = -\frac{b}{a}$$
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Product of roots: $$ x_1 x_2 = \frac{c}{a}$$
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Special Cases:
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Perfect square: Discriminant is a perfect square, rational equal roots
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Incomplete quadratic equation: One variable, forms $$px^2 \pm q =0$$ or $$ax^2 = k, k \neq 0$$
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Applications:
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Solving quadratic inequalities
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Maxima and minima problems
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Sums of squares series
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Problems involving quadratic relationships.
