using the quadratic equation x² - 4x - 5 = 0,
perform the following task:
solve by factoring,
solve by completing the square,
and solve using the quadratic formula
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By factoring:
x² - 4x - 5 = 0
(x - 5)(x + 1) = 0
Setting x - 5 = 0 gives x = 5
Setting x + 1 = 0 gives x = -1
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By completing the square:
Get the constant term, -5, off the left side by
adding +5 to both sides of the equation:
x² - 4x = 5
To the side, multiply the coefficient of x, which is
-4, by 1/2, getting -2. Then square this -2. getting,
(-2)² or 4. Now add 4 to both sides:
x² - 4x + 4 = 5 + 4
The left side will factor as (x - 2)(x - 2) or (x - 2)².
We combine the numbers on the right as 9
(x - 2)² = 9
Now we take square roots of both sides.
x - 2 = ±3
x = 2 ± 3
Using the +, x = 2 + 3, or x = 5
Using the -, x = 2 - 3, or x = -1
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By the quadratic formula:
x² - 4x - 5 = 0
The quadratic formula is:
ax² + bx + c = 0 has solutions:
______
-b ± Öb²-4ac
x = —————————————
2a
In this cases a = 1; b = -4; c = -5
______________
-(-4) ± Ö(-4)²-4(1)(-5)
x = ——————————————————————————
2(1)
_____
4 ± Ö16+20
x = ————————————————
2
__
4 ± Ö36
x = ————————————————
2
4 ± 6
x = —————————
2
Using the +,
4 + 6
x = —————————
2
10
x = ————
2
x = 5
Using the -,
4 - 6
x = —————————
2
-2
x = ————
2
x = -1
Edwin