Explanation

√23 cannot be expressed as a fraction

𝑝/𝑞 since 23 is not a perfect square.

Its decimal expansion is non-terminating and non-repeating, so it is irrational.

Explanation

Trigonometric Identity

Given: cos^2 θ sec^2 θ

Since sec θ = 1/cosθ,

cos^2 θ sec^2 θ = cos^2 θ . 1/cos^2 θ = 1.

Explanation

In algebra, rational fractions are classified into 3 types:

1. Proper fractions → numerator degree < denominator degree.
2. Improper fractions → numerator degree ≥ denominator degree.
3. Mixed fractions → combination of a polynomial and a proper fraction.

Explanation

Number of Solutions

Given: √y+ √y + 3 = 1

Let's solve it:

√y + 3 = 1 - √y

Squaring both sides:

y + 3 = 1 - 2 √y + y

2 = -2 √y

√y = -1

Since √y = 0 for all y = 0, there's no real solution for y.

Explanation

This is from system of linear equations (AX = 0).

If |A| ≠ 0, the system has only the trivial solution (x = 0).

If ∣A∣ = 0, then non-trivial solutions may exist.

Explanation

Additive Inverse

Given: -a - b

The additive inverse of a number x is -x such that x + (-x) = 0.

For -a - b, the additive inverse is -(-a - b) = a + b.

Explanation

The graph of any quadratic function 

y = ax² + bx + c is always a parabola.

It opens upward if a > 0 and downward if a < 0.

Explanation

Complex Conjugate

Given: -6 - i

The conjugate of a complex number a + bi is a - bi.

For -6 - i, a = -6 and b = -1.

The conjugate is -6 + i.

Explanation

Absolute Value Equation

Given: |x + 4| = x

Case 1: x + 4 < 0 or x < -4

$x + 4 = x$

4 = 0, which is not possible.

-(x + 4) = x

-x - 4 = x

-2x = 4

x = -2

Explanation