x8 - 28x5 + 27x2 = 0 x2(x6 - 28x3 + 27) = 0 x2(x3 - 1)(x3 - 27) = 0 x2(x - 1)(x2 + x + 1)(x - 3)(x2 + 3x + 9) = 0 So setting each factor to 0 and solving gives all roots, real and complex. Real: x...

x8 - 28x5 + 27x2 = 0 x2(x6 - 28x3 + 27) = 0 x2(x3 - 1)(x3 - 27) = 0 x2(x - 1)(x2 + x + 1)(x - 3)(x2 + 3x + 9) = 0 So setting each factor to 0 and solving gives all roots, real and complex. Real: x...

We can use the adjugate matrix relation: A adj(A) = In det(A) Then we have: kA adj(kA) = In det(kA) kA adj(kA) = kn In det(A) adj(kA) = kn-1 A-1 det(A) det(adj(kA)) = det(kn-1 A-1 det(A)) det(adj(kA)) = kn(n-1)...

Just let RHS be 0. 2y2 + 3y + 1 = 0 (2y + 1)(y + 1) = 0 y = -1/2, y = -1 We can also see these from the general solution to the ODE. dy/dx = 2y2 + 3y + 1 dy/(2y2 + 3y + 1) =...

Differential Calculus (answer)

To find it by the definition of derivatives, do this: dy/dx = limh→0 [((x+h)2 + 5(x+h) - 2) - (x2 + 5x - 2)]/h = limh→0 [(x2 +(2h+5)x + 5h + h2 - 2) - (x2 + 5x - 2)]/h = limh→0 (2hx + 5h + h2)/h =...

All you have to do is determine what the equation will be in each quadrant when the absolute values are removed. To do this, note that |A| = A for all A ≥ 0, and |A| = -A for all A < 0. You should get the following equations for the quadrants. Q1:...

Calculus Question (answer)

A) x = sin y + cos x 1 = (cos y) dy/dx - sin x (cos y) dy/dx = 1 + sin x dy/dx = (1 + sin x) / cos y B) xy - y3 = sin x y + x dy/dx - 3y2 dy/dx = cos x (x - 3y2) dy/dx = -y +...

Problem 1. Note that the exact root is: x = - 3√4 Machine precision: -1.5874010519681994 Using seed x0 = -2: f(x) = x3 + 4 so f'(x) = 3x2. Thus we iterate: x ← x - (x3 + 4) / (3x2) = 2x/3 - 4/(3x2) x0 = -2 x1...

Triple both sides and then regroup based on pairs which look like a product rule. 3(x2 - 4xy - 2y2)dx + 3(y2 - 4xy - 2x2) dy = 0 3y2 dy - (6y2 dx + 12xy dy) - (12xy dx + 6x2 dy) + 3x2 dx x3 - 6x2y - 6xy2 + y3 = C &n...

simplify the difference quotient (answer)

f(x) = 4 - 3x - x2 [f(x) - f(a)] / (x - a) = [(4 - 3x - x2) - (4 - 3a - a2)] / (x - a) = (-3x - x2 + 3a + a2) / (x - a) = [-3(x - a) - (x + a)(x - a)] / (x - a) = -3 - x - a

algebra 1 help please? (answer)

(z2 - 3z + 2)(z2 + 6z + 8) / (z2 + 3z - 4)(z2 - 4) = (z-1)(z-2)(z+2)(z+4) / (z+3)(z-1)(z+2)(z-2) = (z+4) / (z+3)

You can only give bounds for this number. Let's assume, the n hexagons must form a polyhex (like a polyomino but with regular hexagons). Maximum: The first hexagon contributes 6 vertices and each new hexagon after that contributes at most 4 vertices. MAX...

derivative w(x) = 2cotx sin^2x (answer)

w(x) = 2 cot x sin2 x = 2 cos x sin x = sin 2x dw/dx = 2 cos 2x

probability? (answer)

Coins are memoryless, so it is 1/2.

sin 3x - 1 = 0 sin 3x = 1 3x = 2kπ + π/2 where k ∈ Z x = 2kπ/3 + π/6 where k ∈ Z In the [0,2π), the solutions are π/6, 5π/6, and 3π/2.

x2 + y2 - x + y - 12 = 0 x2 + y2 - 4x - 2 = 0 The easiest way to do this is to subtract the second equation from the first, which yields a linear equation through the two intersection points. 3x + y - 10 = 0 So that means y...

More generally, for odd n, 1n + 2n + 3n + … + nn is a multiple of n. To show that, note that nn is a multiple of n and that we can split the sum into pairs of terms (n-k)n + kn summing to a multiple of n. We...

We can say a couple of things: 1. Since the directrix is vertical and the focus is to it's left, the parabola opens left. 2. We can find the parabola's equation in the form (y - k)2 = 4p(x - h). The vertex is halfway between the focus and directrix, so...

∂f/∂x = 1/(xy) ∂f/∂y = (-1/y2)ln x

Step 1. Use Mark's response to get 2x2 + 5x - 3 = 0 Then use Viete's formulas to get that the sum of the solutions is -5/2. Viete's formulas state that the set R={r1, … ,rn} of roots (real and complex) of anxn + an-1xn-1 + … + a1x + a0 = 0 satisfy...

calculus help (answer)

1. The total area of the squares is 13. The area of the disk is 4π < 4 × 3.25 = 13. Thus clearly, some squares must overlap 2. This is actually a Number Theory problem. If we choose 12 different two digit numbers...