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Lesson 7: Quadratic Equations (III)

February 11, 2011

6. Quadratic equations

 An equation is a quadratic equation in one variable (or unknown) when:

  • It has only one variable or unknown
  • The unknown is squared x2 at least once in the equation.

 For instance: 3x23x=x–1.

 Standard form

 If we bring all the terms over to the RHS (right hand side) and the LHS (left hand side) is equal to 0 we get:

3x2 – 4x + 1 = 0

which is the form we should always use to express quadratic equations to be able to solve them?

 Any quadratic equation in one variable can be expressed using its standard form:

 ax2+bx+c = 0

 where a, b  and c are known numbers and a≠0.

 NOTES:

 1) If a quadratic equation has a = 0, then the equation is a linear equation; this is a first degree equation

 2) If you look at the LHS you find a second degree polynomial in one variable. Because of it, these equations are also named second degree polynomial equations in one variable or second degree equations for shorting (it isn’t very precise).

 3) In many cases once the equation has this form it can be simplified which is very useful.

 7. Solving quadratic equations

 We saw that any quadratic equation can be expressed in its general form ax2+bx+c = 0

 where a, b  and c are known numbers and a≠0.

 We name incomplete quadratic equations to that quadratic equations that have  or . They have the forms:

ax2+c = 0    where    b = 0 

ax2+bx = 0    where    c = 0. 

There are several methods you can use to solve a quadratic equation: 

  • Factoring
  • Completing the Square
  • Quadratic Formula
  • Graphing

 This course, we will practice the Quadratic Formula and a very elemental case of factoring for incomplete quadratic equations.

 First of all, we are going to study the easy cases, when b = 0 or c = 0.

 7.1. Solving quadratic equations of the form ax2+c = 0    where    b = 0 

 In this case you have to follow the same method as in linear equations:

  • Transpose the number to the LHS

 ax2 = c

  • Transpose the coefficient of the quadratic term to the RHS

 x2 = c/a

  • Isolate the unknown by squaring the RHS.

 x  = ±√¯c/a

Take into account that in a square root:

  • If the radicand is positive there are two roots of the same absolute value and opposite signs.
  • If the radicand is zero there one only root, zero.
  • If the radicand is negative there are no real roots.

 When solving quadratic equations of the form  ax2+c = 0    where    b = 0 

  • If –c/a > 0 , there are two solutions x1  = +√¯c/a  and x2  = √¯c/a
  • If –c/a = 0 , there are two solutions x1  = x2  =0
  • If –c/a < 0 , there are no solutions

 7.2. Solving quadratic equations of the form ax2+bx = 0    where    c = 0. 

In this case you have to follow the method of factoring the equation:

In general, if we are solving an equation of the form

  • Factor out the common factor .

 x·(ax + b) = 0

  • Apply the following property: “If the product of two numbers is zero then, one of the factors is zero”.

 x·(ax + b) = 0  →  x = 0    or   ax + b = 0

  • Solve the resulting equations.

 x = 0                    and           ax + b = 0   →   x= –b/a

  When solving quadratic equations of the form  ax2+bx = 0    where    c = 0.the solutions are x = 0 and x= –b/a 

 7.3. Solving equations in the standard form 

 The quadratic formula uses the “a“, “b“, and “c” from “ax2 + bx + c“, where “a“, “b“, and “c” are just numbers; they are the “numerical coefficients”.

 The solutions of an equation of the form  can be obtained by applying the quadratic formula:

 This is there are two solutions corresponding to each sign preceding the square root. We will denote them x1 and x2 .

Practise plugging into the formula on this web page.

Warning: Don ‘t panic if you find “complex numbers” roots with i. These number belong to a set of number you don’t know. If the radicand is negative there are no real solution. This is enough for you.

 The Discriminant

Notice the expression under the radical b2 – 4ac in the quadratic formula is called the discriminant.  From this number we can determine the nature of the solutions of a quadratic equation.

Three possible cases:
1. Δ = b2 – 4ac = 0 Exactly one real number solution exists.
     
2. Δ = b2 – 4ac > 0 Two real number solutions exist.
     
3. Δ = b2 – 4ac < 0 There isn’t any real solution.

 Remark.

 The argument of the square root, the expression b2 – 4ac, is called the “discriminant” because, by using its value, you can discriminate between (tell the differences between) the number of solutions of a quadratic equation.

 No real number has a negative square.

Practise finding the of a quadratic equation and deducing from it the number of solutions of the equation in this web page

 Advises

 1. When you are using the quadratic formula you’re just plugging into a formula. There are no “steps” to remember, and there are fewer opportunities for mistakes, but if you have an incomplete equations use the other techniques, are simpler. You can also use the formula, of course!

 2. When using the formula, make sure you are careful not to omit the “±” sign, and be careful with the fraction line (don’t draw it as being only under the square root; it’s under the initial “–b” part, too). And, though many of your quadratic equations will start with “x2” so a = 1, don’t forget that the denominator of the Formula is “2a“, not just “2”; that is, when the leading term is something like “3x2“, you will need to remember to put the “a = 3″ value in the denominator.

 Finally here you are a video on how to use the discriminant to find out the number of solutions of a quadratic equation.

One comment

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