Explanation

Explanation

Let's break it down step by step:

1. Expand 2x²(1 + 3x):

2x² + 6x³

2. Expand (1 + 4x² - 2x)(5x - 1):

= 5x - 1 + 20x³ - 4x² - 10x² + 2x

= 20x³ - 14x² + 7x - 1

3. Now subtract:

2x² + 6x³ - (20x³ - 14x² + 7x - 1)

= 2x² + 6x³ - 20x³ + 14x² - 7x + 1

= -14x³ + 16x² - 7x + 1

Explanation

${x^}^2+18x+81=0$:

• The first term is x^2, so $a=x$.
• The last term is 81, so ${b^}^2=81$, which means $b=9$ (we take the positive root since the middle term is positive).
• The middle term is 18x. Let's check if it matches 2ab: $2(x)(9)=18x$. It does.

Therefore, the quadratic expression ${x^}^2+18x+81$ can be factored into $(x+9{)^}^2$.

So the factor form of the equation ${x^}^2+18x+81=0$ is $(x+9{)^}^2=0$.

Explanation

Given:
$x^2 - 3x - 10 = 0$

We are told that one factor is $(x - 5)$.
So, we divide the quadratic by $(x - 5)$ to find the other factor:

Method:
Use factoring:

We want two numbers that multiply to $-10$ and add to $-3$.
These numbers are -5 and 2.

Thus,

$$x^2 - 3x - 10 = (x - 5)(x + 2)$$

Explanation

Given:

5x + 8 = 23

Subtract 8 from both sides:

5x = 23 - 8

5x = 15

Divide both sides by 5:

x = 15 / 5

x = 3

Explanation

Given:

a - b = 4

ab = 5

The formula for a³ - b³ is:

a³ - b³ = (a - b)(a² + ab + b²)

We can also use the identity:

a³ - b³ = (a - b)³ + 3ab(a - b)

Substituting the given values:

a³ - b³ = (4)³ + 3 * 5 * 4

= 64 + 60

= 124

Explanation

Given:

1. 6x + 2y = 6300

2. 2x + y = 2600

Multiplying equation (2) by 2:

4x + 2y = 5200

Subtracting:

(6x + 2y) - (4x + 2y) = 6300 - 5200

2x = 1100

x = 550

Explanation

The absolute value of a number is its distance from zero, regardless of direction. Therefore:

|-3| = 3

Explanation

Given:

p^(a - b) × p^(b - c) × p^(c - a)

Using the property of exponents that states a^(m) × a^(n) = a^(m + n), we can simplify:

p^(a - b + b - c + c - a)

= p^(0)

= 1

Explanation

Let's denote the number as x. According to the problem:

133 added to a number (x) is equal to 75 more than 4 times the number (4x + 75)

The correct equation is:

133 + x = 4x + 75

Now, let's solve for x:

133 - 75 = 4x - x

58 = 3x

x = 58 / 3

x = 19.33

The correct equation is indeed:

133 + x = 4x + 75