Algebraic Manipulation | MCQs

Explanation

Explanation

Let's directly calculate:

x - y = 5

x² - y² = 65

(x + y)(x - y) = 65

(x + y)(5) = 65

x + y = 13

Solving these equations:

x - y = 5

x + y = 13

Adding both: 2x = 18

x = 9

y = 4

Let's check:

9 - 4 = 5

9² - 4² = 81 - 16 = 65

Explanation

The original equation is: b - 15 = 32

Solving:

b = 32 + 15 = 47

Explanation

An open sentence contains a variable (like x) and becomes either true or false only when a value is assigned.

x < 3 is open because its truth depends on the value of x.

Explanation

The expression is 2x² + 8x + 8. First, factor out 2:

→ 2(x² + 4x + 4)

Now factor the quadratic:

→ x² + 4x + 4 = (x + 2)(x + 2)

So the full factorization is 2(x + 2)(x + 2).

Explanation

(1+1/x) (1+1/x+1) (1+1/x+2) (1 + 1/x + 3)

Simplify each term:

(x+1)/x * (x+2)/x * (x+3)/x * (x+4)/x

Multiply the numerators and denominators:

(x+1)(x+2)(x+3)(x+4) / x^4

Expand the numerator:

x^4 + 10x^3 + 35x^2 + 50x + 24

Divide by x^4:

1 + 10/x + 35/x^2 + 50/x^3 + 24/x^4

Now, simplify the expression:

Combine like terms:

x + 4/x

Explanation

Explanation

To find the average (arithmetic mean) of a and b, we need to first solve for a and b.

Given equation: 10a + 10b = 35

Divide both sides by 10: a + b = 3.5

Since we want to find the average of a and b, we can use the fact that the average of two numbers is equal to their sum divided by 2.

So, (a + b)/2 = 3.5/2

= 1.75

Therefore, the average (arithmetic mean) of a and b is 1.75.

Explanation

We are given:

4/3​ of a number=22

Let the number be $x$. Then:

4/3​​x=22⇒x=22×3/4​=66/4​=16.5

Now, we need to find:

$$frac{8}{3}x = frac{8}{3} times 16.5 = frac{8 times 33}{6} = frac{264}{6} = 44$$

Explanation

Multiply both sides by 2:

$$|3x| = 6 Rightarrow 3x = ±6 Rightarrow x = ±2$$