Explanation

Let numbers be x and y.

1. Given:

    x + y = 13

    x - y = 23

2. Add both equations:

    2x = 36

    x = 18

3. Put x = 18 in x + y = 13:

    18 + y = 13

    y = -5

4. Product:

    xy = 18 × (-5) = -90

Explanation

Given:

4x/6 = 2

Multiplying both sides by 6/4:

x = 2 × 6/4

x = 12/4

x = 3

Explanation

The numbers 5 and 4 multiply to 20 and add up to 9.

Hence, x² - 9x + 20 = (x - 5) (x - 4)

Explanation

Given: (x + 1/x)(x − 1/x)

Use the formula for difference of squares:  

(a + b)(a − b) = a² − b²

Here, a = x and b = 1/x

So, (x + 1/x)(x − 1/x) = x² − (1/x)²  

= x² − 1/x²

Explanation

We are given the expression:

100² − 99²

This is a difference of squares:

a² − b² = (a − b)(a + b)

So:

100² − 99² = (100 − 99)(100 + 99) = 1 × 199 = 199

Explanation

The standard form of a quadratic equation is ax^2 + bx + c = 0.

It represents a quadratic polynomial set equal to zero, where a, b, and c are constants.

Explanation

Given: y = (2x + 3)⁴  

Using the chain rule:  

dy/dx = 4(2x + 3)³ * d/dx(2x + 3)  

= 4(2x + 3)³ * 2  

= 8(2x + 3)³  

Explanation

We simplify the expression:

x(x⁵)⁴ / x⁴

Step 1: Simplify the exponent  

(x⁵)⁴ = x^(5×4) = x²⁰

Now the expression becomes:  

x × x²⁰ / x⁴

Step 2: Combine the exponents  

Numerator: x × x²⁰ = x²¹  

Now divide: x²¹ / x⁴ = x^(21−4) = x¹⁷

Explanation

We use the algebraic identity:

(a² − b²) = (a − b)(a + b)

Let:  

a = 1,000,001  

b = 999,999

Now:

(1,000,001)² − (999,999)² = (1,000,001 − 999,999)(1,000,001 + 999,999)  

= (2)(2,000,000) = 4,000,000

Explanation

Step 1: Solve for x  

-2x - 5 = 48  

Add 5: -2x = 53  

Divide by -2: x = -26.5

Now find: 2x + 57  

= 2(-26.5) + 57  

= -53 + 57  

= 4