Explanation

Let the two-digit number be 10x + y, where x is the tens digit and y is the unit digit.

Given: x + y = 8

If digits are interchanged, the new number is 10y + x.

Given condition: (10x + y) - (10y + x) = 36

Simplifying: 9x - 9y = 36 → x - y = 4

Solving x + y = 8 and x - y = 4, we get x = 6, y = 2.

The number is 62.

Explanation

Evaluate the polynomial at x = 4

3(4)³ - 2(4)² - 150

= 3(64) - 2(16) - 150

= 192 - 32 - 150

= 10

Explanation

2⅔ can be written as 2^(2/3), which equals ∛(2²) = ∛4 = ³√4.

Explanation

Substitute x = 1 and y = -1 into the expression:

13(x + y) = 13(1 + (-1)) = 13(0).

Therefore, 13(0) = 0.

Explanation

Given:

log₈₁ 9 = x

We can rewrite this in exponential form:

81^x = 9

Since 81 = 9², we can rewrite:

(9²)^x = 9

9^(2x) = 9¹

Equating the exponents:

2x = 1

x = 1/2

x = 0.5

Explanation

r is inversely proportional to p³, so r = k/p³.

Given p = 9 and r = 2:

2 = k/729

k = 2 × 729 = 1458

Now, r = 1/4:

1/4 = 1458/p³

p³ = 1458 × 4 = 5832

p = 18

Explanation

For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.

Given the equation 2x² + kx + 6 = 0, we have:

Sum of the roots = -k/2

Product of the roots = 6/2 = 3

According to the problem, the sum of the roots is equal to four times the product of the roots:

-k/2 = 4 * 3

-k/2 = 12

k = -24

Explanation

The sum of the roots of a quadratic equation ax² + bx + c = 0 is given by -b/a.

Given equation:

x² - 3x - 4 = 0

a = 1

b = -3

Sum of roots = -(-3)/1

= 3

Explanation

To solve the equation:

x​/2−x​/4=2

we can start by finding a common denominator for the terms on the left side.

The common denominator for $frac{x}{2}$ and $frac{x}{4}$is 4, so we rewrite the equation as:

$$frac{2x}{4} - frac{x}{4} = 2$$

Now, combine the fractions on the left side:

2x−x​/4=2

This simplifies to:

x/4​=2

Now, multiply both sides by 4 to isolate $x$

$$x = 8$$

Explanation

The equation is:

(y + 13)(y + a) = 0

Expanding it:

y^2 + ay + 13y + 13a = 0

y^2 + (a + 13)y + 13a = 0

For no linear term, the coefficient of y should be 0:

a + 13 = 0

a = -13