Polynomials(2)

　　Here’s an instance where knowing the perfect square or difference of two square equations can help you:

　　Solve for x: 2x2 + 20x + 50 = 0.

　　To solve this problem by working out the math, you would do the following:

　　If you got to the step where you had 2(x2 + 10x +25) = 0 and realized that you were working with a perfect square of 2(x + 5)2, you could immediately have divided out the 2 from both sides of the equation and seen that the solution to the problem is –5.　　Practice Quadratics　　Since the ability to factor quadratics relies in large part on your ability to “read” the information in the quadratic, the best way to get good is to practice, practice, practice. Just like perfecting a jump shot, repeating the same drill over and over again will make you faster and more accurate. Take a look at the following examples and try to factor them on your own before you peek at the answers.

　　The Quadratic Formula　　Factoring using the reverse-FOIL method is really only practical when the roots are integers. Quadratics, however, can have decimal numbers or fractions as roots. Equations like these can be solved using the quadratic formula. For an equation of the form ax2 + bx + c = 0, the quadratic formula states:

　　Consider the quadratic equation x2 + 5x + 3 = 0. There are no integers with a sum of 5 and product of 3. So, this quadratic can’t be factored, and we must resort to the quadratic equation. We plug the values, a = 1, b = 5, and c = 3 into the formula:

　　The roots of the quadratic are approximately {–4.303, –.697}.　　Finding the Discriminant:　　If you want to find out quickly how many roots an equation has without calculating the entire formula, all you need to find is an equation’s discriminant. The discriminant of a quadratic is the quantity b2 – 4ac. As you can see, this is the radicand in the quadratic equation. If:b2 – 4ac = 0, the quadratic has one real root and is a perfect square. B2 – 4ac > 0, the quadratic has two real roots. B2 – 4ac < 0, the quadratic has no real roots, and two complex roots.　　This information is useful when deciding whether to crank out the quadratic formula on an equation, and it can spare you some unnecessary computation. For example, say you’re trying to solve for the speed of a train in a rate problem, and you find that the discriminant is less than zero. This means that there are no real roots (a train can only travel at speeds that are real numbers), and there is no reason to carry out the quadratic formula.

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