Solving Quadratic Equations

4 Key excerpts on "Solving Quadratic Equations"

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  eBook - ePub

  Years 9 - 10 Maths For Students

  • (Author)
  • 2015(Publication Date)
  • For Dummies

    (Publisher)

  Q uadratic (second-degree) equations are nice to work with because they’re manageable. Finding the solution or deciding whether a solution exists is relatively easy — easy, at least, in the world of mathematics.

  A quadratic equation is a quadratic expression with an equal sign attached. As with linear equations (covered in Chapter 10 ), specific methods or processes, given in detail in this chapter, are employed to successfully solve quadratic equations. The most commonly used technique for solving these equations is factoring, but a quick-and-dirty rule can also be used for one of the special types of quadratic equations. I have to warn you, though, that just because someone puts in some numbers and makes up a quadratic equation, that doesn’t mean there’s necessarily a solution or answer to it. (I show you how to tell if there’s no answer in this chapter.)

  Quadratic equations are important to algebra and many other sciences. Some quadratic equations say that what goes up must come down. Other equations describe the paths that planets and comets take. In all, quadratic equations are fascinating — and just dandy to work with.

  Squaring Up to Quadratics

  A quadratic equation contains a variable term with an exponent of 2 and no variable term with a higher power.

  A quadratic equation has a general form that goes like this: ax2 + bx + c = 0. The constants a, b and c in the equation are real numbers, and a cannot be equal to 0. (If a were 0, you wouldn’t have a quadratic equation anymore.)

  If the equation looks familiar, it means that you’ve read Chapter 9 , which talks about factoring and working with quadratic expressions. Remember: An expression consists of one or more terms but has no equal sign. Adding an equal sign changes the whole picture: Now you have an equation that says something. The equation forms a true statement if the solutions are put in for the variables.

  Here are some examples of quadratic equations and their solutions:

  • 4x2 + 5x − 6 = 0: In this equation, none of the coefficients is 0. The two solutions are x = −2 and

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  eBook - ePub

  Foundations of GMAT Math

  • Manhattan GMAT(Author)
  • 2011(Publication Date)
  • Manhattan Prep Publishing

    (Publisher)

  Chapter 7: Quadratic Equations

  In This Chapter:

    •  Manipulating quadratic expressions and Solving Quadratic Equations Mechanics of Quadratic Equations In high school algebra, you learned a number of skills for dealing with quadratic equations. For the GMAT, you need to relearn those skills.

  Let's define terms first. A quadratic expression contains a squared variable, such as x2 , and no higher power. The word “quadratic” comes from the Latin word for “square.” Here are a few quadratic expressions:

  z2

  y2 + y – 6

  x2 + 8x + 16

  w2 – 9

  A quadratic expression can also be disguised. You might not see the squared exponent on the variable explicitly. Here are some disguised quadratic expressions.

  z × z

  (y + 3)(y – 2)

  (x + 4)2

  (w – 3)(w + 3)

  If you multiply these expressions out—that is, if you distribute them—then you see the exponents on the variables. Note that the second list corresponds to the first list exactly.

  A quadratic equation contains a quadratic expression and an equals sign.

  Quadratic expression = something else

  A quadratic equation usually has two solutions. That is, in most cases, two different values of the variable each make the equation true. Solving a quadratic equation means finding those values.

  Before you can solve quadratic equations, you have to be able to distribute and factor quadratic expressions.

  Distribute (a + b)(x + y) Use FOIL

  Recall that distributing means applying multiplication across a sum.

  =	5 × 3	+	5 × 4
  equals	five times three	plus	five times four.

  You can omit the multiplication sign next to parentheses. Also, the order of the product doesn't matter, and subtraction works the same way as addition. Here are more examples:

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  eBook - ePub

  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  in order to write the solution:

  If we allow the usual algebraic processes for taking the square roots of fractions, that is, then the solution simplifies to

  These numbers involving are called complex numbers. A convenient way to write complex numbers is to use the notation Then, we can write and so the solution above can be expressed as

  Complex numbers and the notation i will be explained further in section 3.6 .

  We now derive the general formula for the solution of a quadratic equation, known as the quadratic formula. In order to solve the equation ax2 + bx + c = 0, we replace the left-hand side of the equation with the expression from formula (3.2). Thus, we need to solve the equation

  As in the examples above, this means that The more usual way to write this is to use the ± notation, which means + or −, that is,

  The quadratic formula provides a shortcut for solving a quadratic equation. All one has to do is identify the values of a, b, and c with the coefficient of x2 , the coefficient of x and the constant term, respectively, and substitute them into the quadratic formula.

  REMARK 3.3.1. The quantity b2 − 4ac that appears inside the square root in formula (3.3) is called the discriminant. More about this later.

  EXAMPLE 3.3.5. We can solve the equation −4x2 − 13x + 7 = 0 by substituting a = −4, b = −13, and c = 7 in the quadratic formula, that is,

  3.4POLYNOMIALS

   

  Polynomials and the algebraic operations that can be applied to them are the main concern of this chapter. We will begin with the formal definition of a polynomial (in one variable) and then explain how polynomials can be added, subtracted, multiplied, and divided.

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  eBook - ePub

  The Mathematics That Every Secondary School Math Teacher Needs to Know

  • Alan Sultan, Alice F. Artzt(Authors)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  Let us now illustrate a typical secondary school problem where a quadratic equation is solved by completing the square. Notice that this method requires that a, the coefficient of x 2, be equal to 1. Example 3.15 Using the method of completing the square, solve the equation x 2 + 6 x + 1 = 0. Solution : We subtract 1 from each side of the equation to get x 2 + 6 x = −1. We complete the square on the left side by adding 9. Of course to keep the equation balanced, we need to do the same to the right hand side. Our equation becomes: x 2 + 6 x + 9 = −1 + 9. This is the same as (x + 3) 2 = 8. Thinking of (x + 3) as y, this tells us we have y 2 = 8 and hence y = ± 8. Replacing y by x + 3 we have, x + 3 = ± 8. So x = - 3 ± 8. Example 3.16 Use the method of completing the square to solve the equation 3 x 2 + 4 x − 2 = 0. Solution : We add 2 to both sides to get 3 x 2 + 4 x = 2. To use the method of completing the square, we need the coefficient of x 2 to be 1. So we divide the equation by 3 to get x 2 + 4 3 x = 2 3. We add [ 1 2 (4 3) ] 2 = 4 9 to both sides of the equation to get x 2 + 4 3 x + 4 9 = 2 3 + 4 9 which just becomes x + 2 3 2 = 10 9. From this we get that (x + 2 3) = ± 10 9, and so, x = - 2 3 ± 10 9. It is exactly in this way that we derive the quadratic formula. Here it is for completeness. Example 3.17 Derive the quadratic formula. Solution : We start with the equation ax 2 + bx + c = 0 where a > 0. We then subtract c from both sides to get a x 2 + b x = - c. Since we need the coefficient of x 2 to be 1, we divide both sides by a to get x 2 + b a x = - c a. We complete the square on the left side by adding (1 2 ⋅ b a) 2, or just b 2 4 a 2. We get x 2 + b a x + b 2 4 a 2 = - c a + b 2 4 a 2. (3.9) Now the left side of equation (3.9) is a perfect square,. the square of (x + b 2 a). Thus, we have (x + b 2 a) 2 = - c a + b 2 4 a 2 which can be rewritten as (x + b 2 a) 2 = - 4 a c 4 a 2 + b 2 4 a 2. Combining the two fractions on the right we

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