find the root of the equation 2x square -x+ 1/3equal to 0
Answers
Equations
Any equation in the form
ax2 + bx + c = 0 with a not equal 0
is called a quadratic equation!
A value that satisfies this equation is called a root, zero or solution of the equation!! We have three methods we can use to solve these equations:
1) factoring (easy)
2) completing the square
3) quadratic formula
Factoring review!
Solve each by factoring
1) x2 + 4x -5 = 0
Solution: factor (x + 5)(x - 1) = 0
Therefore, x = -5 or x = 1
2) (3x - 2)((x + 4) = -11
Solution: Foil first 3x2 + 10x - 8 = -11
Put in correct form 3x2 + 10x + 3 = 0
factor (3x + 1)(x + 3) = 0
Solution: x = -1/3 or x = -3
Completing the square!!
Follow the explanation and sample problem to review completing the square
1) Use completing the square to find the solutions for:
2x2 - 12x - 9 = 0
Solution:
Move the constant to the other side: 2x2 - 12x = 9
Divide by the coefficient of x2 x2 - 6x = 9/2
Take half the coefficient of x and square: x2 - 6x + 9 = 9/2 + 9
Factor the trinomial square: (x - 3)2 = 27/2
Take the square root of both sides:
Simplify the radical:
Isolate for x:
Quadratic Formula
As proved in class the quadratice formula is derived by completing the square. Here is the formula:
If ax2 + bx + c = 0 then the roots of the equation are:
Look familiar? It better!!
Solve the following problem by using the quadratic formula.
2x2 + 5 = 3x
2x2 - 3x + 5 = 0 (putting in correct form)
a = 2, b = -3 and c = 5 Use the formula:
The Discriminant!
The quantity under the radical in the quadratic formula can tell us alot about the nature of the solutions. Therefore, it is given a special name. The discriminant is b2 - 4ac
If the discriminant is less than zero, then you will be taking the square root of a negative number yielding complex solutions. If the discriminant equals zero, you have one real solution (namely -b/2a) (Where have I heard that before?). If the discriminant is greater than zero, then we have two different real solutions. This is summarized in the following chart:
discriminant Types of solutions
b2 - 4ac < 0 Two complex conjugates
b2 -4ac = 0 One real (double root)
b2 - 4ac > 0 Two different real roots
Warning! Warning! Danger Ahead!!
Be extremely careful in solving problems like this:
x2 = x
It is very tempting to divide both sides by x to get:
x = 1
This is incorrect, because you have completely eliminated one of the solutions. Never divide both sides of an equation by a variable if it cancels from both sides!!
Correct way to do the problem is as follows:
x2 = x
x2 - x = 0
x(x - 1) = 0
Solutions are x = 0 and x = 1
Honk! Honk! That about does it for this section! Onward and upward!
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