Here we're presented with a quadratic equation which needs to be expanded and then rewritten in descending powers of x:
1x^2 + m^2x^2 + 6mx + 9 - 3 = 0.
Let's group like terms: 1x^2 + m^2x^2 + 6mx + 6 = 0.
The first 2 terms can be rewritten as a single term: (1+m^2)x^2, and so we now have:
(1+m^2)x^2 + 6mx + 6 = 0.
We must now calculate the discriminant and set the resulting expression = to 0, as a preliminary to finding the value of m for which the given quadratic has equal roots:
discriminant: (6m)^2 - 4(1+m^2)(6) = 0
Then 36m^2 - 24(1+m^2) = 0, which simplifies to 12m^2 - 24 = 0.
Then 12 m^2 = 24; m^2 = 2, and m = √2.
When the discriminant is zero, as it is here when m = √2, then the given quadratic has two equal roots.
Answer:
Step-by-step explanation:
The perimeter of ABCD is 30cm
The Sum of the perimeter of three rectangles at the corner is 20cm.
following the variables as the same as you assigned
large rectangle: u by v
rectangle center A: 2a by 2b
rectangle center B: 2c by 2d
rectangle center D: 2e by 2f
2 equations (from the givens):
2u+2v=30
2(a + b + c + d + e + f )= 20
now to find the length
a+b+a/2+b/2+(v-b/2-d/2)+c/2+d+c+d/2+(u-c/2)+(v-f/2)+e/2+f+e+f/2+(u-e/2-a/2)
which is (a+b+c+d+e+f)+u+v+u+v = 10+30=40
As the three longitudinal lines are parallel:
angle 4 = 5 = 6 = 82 degree (corresponding angles) = 9 = 8 = 7 = 82 degrees (vertical angles)
and
angle 11 = 180-82= 98 degrees (linear pair)
so:
angle 10 = 11 = 12 = 98 degrees (corresponding angles) = 3 = 2 = 1 = 98 degrees (vertical angles)
.
hope you got it
X = amount of 90% alloy
y = amount of 70% alloy
x + y = 60
0.9x + 0.7y = 0.85*60
0.9x + 0.7(60 - x) = 0.85*60
(0.9 - 0.7)x = (0.85 - 0.7)*60
x = (0.85 - 0.7)*60/(0.9 - 0.7)
x = 45 ounces
y = 60 - 45
y = 15 ounces