Quadratic Equations

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A quadratic equation in variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. This is called the standard form of a quadratic equation. The term 'quadratic'…

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If a = 0, then the equation becomes bx + c = 0, which is a linear equation (degree 1), not a quadratic equation (degree 2). The term ax² is essential to make it quadratic, so a ≠ 0 is a necessary cond…

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A real number α is called a root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0. In other words, when we substitute x = α in the equation, the left-hand side equals zero. The roots are…

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The zeros of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are exactly the same. If α is a zero of the polynomial, then α is also a root of the equatio…

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A quadratic equation can have at most 2 roots. This is because a quadratic polynomial has degree 2, and a polynomial of degree n can have at most n zeros. Since the zeros of the polynomial are the sam…

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x² - 5x + 6 = 0 We need two numbers that multiply to 6 and add to -5: -2 and -3 x² - 2x - 3x + 6 = 0 x(x - 2) - 3(x - 2) = 0 (x - 2)(x - 3) = 0 Therefore: x - 2 = 0 or x - 3 = 0 So x = 2 or x = 3…

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Completing the square is a method where we convert ax² + bx + c = 0 into the form a(x + p)² + q = 0 by adding and subtracting appropriate terms. It's useful when factorization is difficult or when we …

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The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a), where a ≠ 0 and b² - 4ac ≥ 0. This formula gives the roots of any quadratic equation and is derived using the method of completing the square.