§8.1 - Solving Quadratic Equations
Notes
Factoring may be used to solve quadratic equations. When it does, the
solutions are rational.
Examples: Solve by factoring.
(A) x 2 + 5x = 24
(B) 3x 2 = 4 − 11x
(C) 9x 2 + 12 = 3 + 12x + 5x 2
Exercise 1: x 2 − 15x + 54 = 0
(Math 1010)
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§8.1 - The Square Root Property
Notes
Using the square root property may solve a quadratic equation. When it
does, the solutions may be irrational or rational.
The Square Root Property: Suppose u is an algebraic
expression
and
√
√
d > 0. Solving u 2 − d = 0√by factoring gives (u + d)(u − d) = 0
whose solutions are u = ± d.
Examples: Solve using the square root property.
(A) 3x 2 = 15
(B) (x − 2)2 = 10
(C) (3x − 6)2 − 8 = 0
(Math 1010)
Exercise 23: 6x 2 = 54
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§8.1 - The Square Root Property
Notes
When u is an algebraic expression and d < 0, the quadratic equation
u 2 − d = 0 will have complex solutions. That is,
p
When d < 0, u 2 − d = 0 has solutions u = ± |d|i
Examples: Solve using the square root property.
(A) x 2 + 8 = 0
(B) (x − 4)2 = −3
(C) 2(3x − 5)2 + 32 = 0
Exercise 45: x 2 + 4 = 0
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§8.1 - Special Quadratic Forms
Notes
The special form to discuss last is an equation in ”quadratic form.”
Suppose u is an algebraic expression. Then we can use our techniques to
solve in terms of u:
au 2 + bu + c = 0
Examples: Identify the algebraic expression u.
(A) x 4 − 13x 2 + 36 = 0
(B) x 4 − 5x 2 + 4 = 0
√
(C) x − 5 x + 6 = 0
(Math 1010)
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§8.1 - Special Quadratic Forms
Notes
The special form to discuss last is an equation in ”quadratic form.”
Suppose u is an algebraic expression. Then we can use our techniques to
solve in terms of u:
au 2 + bu + c = 0
Next we solve the quadratic equation in terms of u, then solve u in terms
of x.
Examples: Write each in terms of u, solve for u, then solve for x.
(A) x 4 − 13x 2 + 36 = 0
(B) x 4 − 5x 2 + 4 = 0
√
(C) x − 5 x + 6 = 0
(Math 1010)
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Notes