Transcript Section 1-6 - MrsBarnesTrig

Section 1-6
Solving Quadratic Equations
Quadratic Equation
• Any equation that can be written in the
form ax 2  bx  c  0 where a ≠ 0, is called a
quadratic equation.
Roots
• A root, or solution, of a quadratic equation
is a value of the variable that satisfies the
equation.
Solving Quadratic Equations
• Three methods for solving quadratic
equations are:
1 – factoring
2 – completing the square
3 – the quadratic formula.
Factoring
• Whenever the product of two factors is
zero, at least one of the factors must be
zero. For example, if (3x – 2)(x + 4) = 0
then 3x - 2 = 0 or x + 4 = 0. A quadratic
equation must be written in the standard
form before it can be solved by factoring.
Factoring Review
• Factor:
x  3x
2
x  4x  5
2
6 x  5x  6
2
12  5 x  2 x
x  8 x  16
2
x 4
2
2
Completing the Square
• The method of transforming a quadratic
equation so that one side is a perfect
square trinomial is called completing the
square.
Completing the Square
• Step 1: Divide both sides by the coefficient of
so that will have a coefficient of 1.
• Step 2: Subtract the constant term from both
sides.
• Step 3: Complete the square. Add the square
of one half the coefficient of x to both sides.
• Step 4: Take the square root of both sides and
solve for x.
The Quadratic Formula
• The quadratic formula is derived by
completing the square.
• The roots of the quadratic equation
ax 2  bx  c  0 are given by:
(a ≠ 0)
 b  b 2  4ac
x
2a
The Discriminant
• The quantity under the radical sign in the
quadratic formula, b 2  4ac , can tell you
whether the roots of a quadratic equation
are real or imaginary. Because of this
“discriminating ability” is called the
discriminant.
The Nature of the Discriminant
• Given the quadratic equation , where a, b,
and c are real numbers.
• If b  4ac < 0 there are two conjugate
imaginary roots
• If b  4ac = 0 there is one real root (called a
double root)
• If b  4ac > 0 there are two different real
roots
2
2
2
• In the special case where the equation
ax 2  bx  c  0 has integral coefficients
2
b
and  4ac is the square of an integer, the
equation has rational roots. That is, if b 2  4ac
is the square of an integer, then ax 2  bx  c  0
has factors with integral coefficients.
Choosing a Method of Solution
• You can always use the quadratic formula
to solve any quadratic equation.
• But if a, b, and c are integers and is a
perfect square use the factoring method.
If the equation has the form
x 2 + (even number)x + constant = 0 use
the method completing the square.
Losing or Gaining a Root
• If an equation contains variables on both
sides or variables in a denominator, then
you must carefully organize your method
for solving in order not to lose a root or
gain a root.
Losing a Root
• It is possible to lose a root by dividing both
sides of an equation by a common factor.
• Examine incorrect method on p. 32. You
need to keep in mind that when you divide
both sides of the equation by x – 1, 1 is a
root.
• Examine correct methods on p. 33
Gaining a Root
• It is possible to gain a root by
squaring both sides of an equation.
Another way to gain a root is by
multiplying both sides of an equation
by an expression. Any gained root
called an extraneous root, satisfies
the transformed equation, but not the
original equation. CHECK YOUR
SOLUTIONS IN THE ORIGINAL
EQUATION TO ELIMINATE
EXTRANEOUS ROOTS.
Example
• Solve: 2 x  1  4
2
Example
• Solve: y  8 y  2
2
Example
• Solve:
3z  6  2 z  1