When roots of polynomials occur in radical form, they occur as two conjugates.
That is,
The conjugate of (a + √b) is (a - √b) and vice versa.
To show that the given conjugates come from a polynomial, we should create the polynomial from the given factors.
The first factor is x - (a + √b).
The second factor is x - (a - √b).
The polynomial is
f(x) = [x - (a + √b)]*[x - (a - √b)]
= x² - x(a - √b) - x(a + √b) + (a + √b)(a - √b)
= x² - 2ax + x√b - x√b + a² - b
= x² - 2ax + a² - b
This is a quadratic polynomial, as expected.
If you solve the quadratic equation x² - 2ax + a² - b = 0 with the quadratic formula, it should yield the pair of conjugate radical roots.
x = (1/2) [ 2a +/- √(4a² - 4(a² - b)]
= a +/- (1/2)*√(4b)
= a +/- √b
x = a + √b, or x = a - √b, as expected.
Answer:
Length: 14 feet Width: 1 foot
Step-by-step explanation:
This is the only way to have a rectangle that has a perimeter of 30 feet, and an area of 14 feet.
For perimeter:
14 + 14 = 28
1 + 1 = 2
2 + 28 = 30
These dimensions have a perimeter of 30.
For area:
14 x 1 = 14
These dimesnions also have an area of 14.
So, 14 and 1 are the correct dimensions!
Area of a triangle = (1/2)*base*height
For both of the triangles, you have the base (8.8 for the triangle on the left, 7.6 for the triangle on the right) and the side lengths, but not the height. But since both are isosceles triangles, you can find the height using the pythagorean theorem.
5.
First divide the triangle vertically into two triangles (see attached picture). Now you have two right triangles, you can apply the pythagorean theorem on either one of them to find the height. The pythagorean theorem says that for a right triangle, 
, where c is the hypotenuse and a and b are the sides of the triangle.
Substituting the given values and rounding to nearest tenth:
Now that you have the height, you can find the area of the entire triangle.
A = (1/2)*base*height
A = (1/2)*8.8*9.0 = 39.6
6.
Same procedure.
A = (1/2)*base*height
A = (1/2)*7.6*9.2 = 35.0
Answer:
90%
Step-by-step explanation:
