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Chapter 4: Quadratic Equations

Roots of Even Exponents   

    In previous discussions, we have talked about the polynomials. There is a special kind of polynomial called the Quadratic Equation. The equation ax² + bx + c = 0, a ¹ 0, is called the general quadratic equation in x. A quadratic equation with no x-term is called a pure quadratic equation. Quadratic equations are to the second powers. So we feel it’s necessary to talk about the second power or other even powers.

    When you take the square roots of a second power number or other even power numbers, you always get two answers: the positive ones and the negative ones. This is called the Square Root Property of Equations. It states that for all real numbers m and n, n ³ 0, if m² = n, then m = Ö n or m = - Ö n. When comes to solve a quadratic formula, you need to use the quadratic formulas. Although you can solve the quadratic equations by completing the square, sometimes the square is not perfect, and you can’t solve it. The quadratic formula can be used on all the quadratic equations. Quadratic formula states that the roots of ax²+bx+c=0, a¹ 0, are [–b+Ö (b²-4ac)]/2a and [–b-Ö (b²-4ac)]/2a.

    The part of the quadratic formula under the radical sign is called the discriminant. We can use the discriminant to determine the outcomes of the roots of the quadratic equations. If the discriminant is equal to zero, the roots of the quadratic equation are equal to each other and rational. If the discriminant is larger than zero and is a perfect square, the roots of the quadratic equation are unequal and rational. If the discriminant is larger than zero and is not a perfect square, the roots of the quadratic equaiton are unequal and irrational. If the discriminant is smaller than zero, the roots of the quadratic equation are unequal and not real.

Roots of Negative Numbers

    The most time we take a square roots of a number, we do it with positive numbers. But what about the negative numbers. Can we take the square roots of a negative number. The answer is yes, and this comes to the concept of complex numbers. A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit. Ö -1 = i and i² = -1. Here are some additional properties of complex numbers:

  Square Root of a Negative Number: If a is a positive real number, then Ö -a = iÖ a.

  Equality of Complex Numbers: For all complex numbers a + bi and c + di, a + bi = c + di if and only if a = c and b

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