Download Math 152B – Cass Gustafson – 8.3 The Discriminant and Equations

Math 152B – Cass
Gustafson – 8.3
The Discriminant and Equations That Can Be Written in Quadratic Form
I.
The Discriminant
A. The radicand b2 – 4ac in the quadratic formula is called the discriminant.
B. It is called the discriminant because:
Discriminant
b2 – 4ac
Solutions to ax2 + bx + c = 0
2
The equation has two unequal REAL solutions.
2
If b – 4ac is a perfect square, the solutions are RATIONAL numbers.
2
If b – 4ac is NOT a perfect square, the solutions are IRRATIONAL CONJUGATES.
2
The equation has two equal REAL solutions, called a double root and if a, b, and c are
rational, it would be a RATIONAL number.
2
The equation has NO REAL solutions. It has two IMAGINARY solutions. They would be
COMPLEX CONJUGATES.
If b – 4ac > 0
If b – 4ac = 0
If b – 4ac < 0
Ex. 1: For each equation, compute the discriminant. Then determine the number and type of
solutions. Do not solve the equations.
a) 2x2 – 7x – 4 = 0
b) 3x2 – 2x + 4 = 0
Ex. 2: Find the value of k that will make the solutions of kx2 – 20x + 25 = 0 equal.
II. Determining Which Method to Use: What method do we use to solve a quadratic equation?
Form of the Quadratic Equation
Most Efficient Solution Method
ax2 + bx + c = 0 and ax2 + bx + c can be factored easily
Solve by factoring.
ax2 + c = 0 (The equation has no “x” term, b = 0)
Solve using the square root property
(a polynomial)2 = a number
Solve using the square root property
2
Solve by completing the square or by
using the quadratic formula.
2
ax + bx = c = 0 and ax + bx + c
cannot be factored or is too difficult to factor
x =
−b ±
b 2 − 4ac
2a
III. Solving Equations That Can Be Written in Quadratic Form
When equations are quadratic in form, they will be of the pattern au2 + bu + c = 0 where u can be
any algebraic expression. Some examples follow:
Example
(x
2
Quadratic-Like
x 4 − 9x 2 + 8 = 0
(x )
x − 3 x − 4=0
( x)
)
−1
2
(
)
− x2 − 1 − 2 = 0
m−2 − 6m−1 + 4 = 0
2
n5 − n
1
5
− 2=0
2
(x
2
2
2
)
−1
(m )
−1
1
5
− 3
2
2
( )
n
( )
− 9 x2
In General
u2 – 9u + 8 = 0
+ 8=0
( x) −
(
)
u2 – u – 2 = 0
− x2 − 1 − 2 = 0
(
)
− 6 m−1 + 4 = 0
2
u2 – 3u – 4 = 0
4=0
( )
1
− n5
u2 – 6u + 4 = 0
u2 – u – 2 = 0
− 2=0
What is “u” in each of the examples above?
Ex. 3: Solve the following equations. Notice that sometimes we must check our solutions!
a) x +
Gustafson – 8.3
x − 12 = 0
b) x
2
3
= − 3x
1
3
+ 10
2
c)
12
6
+
= 5
x
x+3
d) 28x −2 − 3x −1 − 1 = 0
IV. Solving Equations for a Specified Variable
Ex. 4: Solve each equation for the indicated variable.
a) a2 + b2 = c2 for a
b) kx = ay – x2 for x
V. Verifying Solutions of a Quadratic Equation by Calculating the Sum and Product
This subsection is OPTIONAL. Solve problems 63-69 (odd), but don’t do the verification.
Consider the solutions of a quadratic equation based on the quadratic formula …
r1 =
−b + b2 − 4ac
2a
r2 =
−b − b2 − 4ac
2a
Calculate the sum …
Calculate the product …
Gustafson – 8.3
3
We can use this to state that if
r1 + r2 = −
b
a
and
r1 r2 =
c
a
then r1 and r2 are solutions of ax2 + bx + c = 0. We can also use these formulas to check our
answers.
Ex. 5: Solve 8x2 – 2x – 3 = 0 and verify that the sum of the solutions is −
the solutions is
Gustafson – 8.3
b
and that the product of
a
c
.
a
4