Explanation

 a² - b²  = (a+b) (a-b)

Explanation

Let the number of sweets Roni gives to Tony be x.

After giving x sweets: Roni will have 2 25−x sweets.

Tony will have 55+x sweets.

We want Tony to have four times as many sweets as Roni, so: 55 + x = 4 ( 25 − x )

1. Expand the equation:

   $$55 + x = 100 - 4x$$
   

2. Add $4x$ to both sides to combine the $x$-terms:

   $$55 + 5x = 100$$
   

3. Subtract 5 from both sides:

   $$\mathrm{<font\; color="\#0000ff">5</font>}\mathrm{<font\; color="\#0000ff">x</font>}<font\; color="\#0000ff">=</font>\mathrm{<font\; color="\#0000ff">45</font>}$$5x = 45
   

4. Solve for $x$:

   $$$$x=45/5=9x 
   

Explanation

$$lim_{x to 0} frac{1 - cos x}{x^2} = lim_{x to 0} frac{sin x}{2x} = frac{1}{2}$$

Explanation

What is the value of the integral:

∫e^x2+lnx dx

Step 1: Simplify the integrand

Recall:

e^a+b=e^a⋅e^b

So,

e^x^2+lnx=e^x^2⋅e^lnx=x⋅e^x^2

Step 2: The integral becomes:

∫x⋅e^x^2 dx

Now use substitution:

Let $u = x^2 Rightarrow du = 2x , dx Rightarrow frac{1}{2} du = x , dx$

So,

$$int x cdot e^{x^2} , dx = frac{1}{2} int e^u , du = frac{1}{2} e^u + C = frac{1}{2} e^{x^2} + C$$

=1​/2∫e^u du

=1​/2e^u+C

=1​/2ex^2 +C

Explanation

Explanation

1. $$$$32−1=31

Explanation

A(b+c) =  (ab + ac)

Explanation

Let total participant =X then X/4+ X/6+ X/8 + 33= X

LCM of 4,6,8 = 24 , multiply each by 24 

X/4*24+ X/6*24+ X/8 *24+ 33(24) = 24X

6x+4x+3x+792=24x

792=24x+13x

11x=792

x=792/11=72

Explanation

i^25 = i^(24+1) = i^1 = i (since i^4 = 1)

(1/i)^32 = (i^-1)^32 = i^(-32) = i^0 = 1 (since i^4 = 1)

So, (i^25 - (1/i)^32)^2

= (i - 1)^2

= i^2 - 2i + 1

= -1 - 2i + 1

= -2i

Explanation