Help explain these quadratic(?) equations please
[From: ] [author: ] [Date: 11-12-21] [Hit: ]
By indicating the sign behavior of each factor in these 3 regions you can determine the sign behavior of their quotient.[Notice that x=-1 is NOT in the domain.Division by zero is undefined.]You find then that the solution interval is (-1, 5] or written another way:-12)x - 3 = 10/xMult by x and this becomes a rather conventional quadratic.[Domain does NOT include zero.......
then apply the quadratic formula and get (1± √97)/2 for the paren-quantity. Then solve for x (by adding -54 to each side) giving x=(-107± √97)/2.
4)(x - 5)/(x+1) less than or = to 0 The zeros of the factors divide the number line into 3 regions. By indicating the sign behavior of each factor in these 3 regions you can determine the sign behavior of their quotient. [Notice that x=-1 is NOT in the domain. Division by zero is undefined.] You find then that the solution interval is (-1, 5] or written another way: -1
2) x - 3 = 10/x Mult by x and this becomes a rather conventional quadratic. [Domain does NOT include zero.]
3) x - 2 (√x) - 24= 0 This one is quite similar to #1. Treat the parentheses quantity as a new variable (and I think that this one even factors! --no having to use quadratic formula.) Note that you cannot have solutions that are less than zero (because there are no real sq roots for negative number.
-
1.) (x + 54)² - (x + 54) - 24 = 0
x² + 108x + 2916 - x - 54 - 24 = 0
x² + 107x + 2838 = 0
x² + 107x = - 2838
x² + 107x + 2862.25 = - 2838 + 2862.5
(x + 53.5)² = 24.5
x + 53.5 = √24.5
x + 53.5 = ± 4.9497
x = - 53.5 ± 4.9497
If x = - 53.5 + 4.9497,
x = - 48.55
If x = - 53.5 - 4.9497,
x = - 58.45
x { - 58.45, - 48.55 }
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
2.) x - 3 = 10 / x
x(x - 3) = x(10 / x)
x² - 3x = 10
x² - 3x - 10 = 0
(x - 5)(x + 2) = 0
If the product of two terms equals zero, then one or both terms equal zero.
If x - 5 = 0,
x = 5
If x + 2 = 0,
x = - 2
x { - 2, 5 }
¯¯¯¯¯¯¯¯
3.) x - 2√x - 24 = 0
2√x = x - 24
4x = (x - 24)²
4x = x² - 48x + 576
x² - 48x - 4x + 576 = 0
x² - 52x + 576 = 0
(x - 36)(x - 16) = 0
If x - 36 = 0,
x = 36
If x - 16 = 0,
x = 16
x { 16, 36 }
¯¯¯¯¯¯¯¯¯¯
4.) (x - 5) / (x + 1) ≤ 0
If a fraction is negative, then either the numerator or the denominator is negative, but not both. The denominator cannot equal zero, lest division by zero occurs, which is undefined, so
If x - 5 ≤ 0,
x ≤ 5
¯¯¯¯¯
If x + 1 < 0,
x < - 1
¯¯¯¯¯¯
-
(x+54)(x+54)-(x+54)-24=0
x^2+108x+2916-x-54-24=0
x^2+107x-2838=0
Using quadratic formula, x=( -107+ or - sqr (107^2-4(-2838(1))) /2
x=22 or -129
x-3=10/x
Clearing fractions by multiplying through by x, then rearranging, we get
x^2-3x-10=0
Using quadratic formula,
x=5 or -2
-
x-2√x-24=0
-2√x=x+24
-2x=x^2+576
x^2+2x+576=0
Use quadratic formula:
(-b+/-√b^2-4ac)/2a; ax^2+bx+c=0
keywords: please,equations,explain,Help,these,quadratic,Help explain these quadratic(?) equations please
