Explanation

(x + 3)(x + k) = x^2 + kx + 3x + 3k

= x^2 + (k + 3)x + 3k

No linear term means k + 3 = 0

So, k = -3

Explanation

(x, y) is equidistant from (0, 1) and (3, 3).

Distance formula gives 6x + 4y = 17.

Check (1.5, 2): 6(1.5) + 4(2) = 9 + 8 = 17.

So, (1.5, 2) is the point.

Explanation

x - 1/x = 4

Square it:

x² - 2 1/x² = 16

Add 2:

x² + 1/x² = 18

Explanation

r is inversely proportional to p³, so r = k/p³.

Given p = 9 and r = 2:

2 = k/729

k = 2 × 729 = 1458

Now, r = 1/4:

1/4 = 1458/p³

p³ = 1458 × 4 = 5832

p = 18

Explanation

The discriminant D = b^2 - 4ac = 1^2 - 4(4)(1) = 1 - 16 = -15.

Since D = -15 < 0, the roots are Imaginary and unequal.

Explanation

Calculate the cube root of 32768.

³√32768 = ³√32³ = 32

Explanation

Given log₃x = -2

This can be rewritten in exponential form:

3^(-2) = x

x = 1 / 3²

x = 1 / 9

Explanation

Factorize the quadratic expression.

3x^2 - 9x + 6 = 3(x^2 - 3x + 2)

= 3(x^2 - 2x - x + 2)

= 3(x(x - 2) - 1(x - 2))

= 3(x - 1)(x - 2)

Explanation

(x - y)^2 = x^2 - 2xy + y^2 and (x + y)^2 = x^2 + 2xy + y^2.

Subtracting these, we get: x^2 - 2xy + y^2 - (x^2 + 2xy + y^2) = x^2 - 2xy + y^2 - x^2 - 2xy - y^2 = -4xy.

Explanation

For the quadratic equation 3x^2 - 6x + k = 0 to have equal roots, the discriminant b^2 - 4ac must be 0.

Here, a = 3, b = -6, and c = k.

Setting the discriminant to 0 gives (-6)^2 - 4(3)(k) = 0.

Solving this, 36 - 12k = 0, which yields k = 3.