Explanation

Area of triangle = ½ × base × height

If base becomes 2b and height becomes 2h:

New area = ½ × (2b) × (2h)

= 4 × (½ × b × h)

So, area becomes 4 times the original.

Last verified on 15-05-2026

Explanation

Given:

a² + b² = 65

a = 1

Put value of a:

1² + b² = 65

1 + b² = 65

Subtract 1 from both sides:

b² = 64

Take square root:

b = 8 (positive value)

So, b = 8.

Last verified on 15-05-2026

Explanation

• A scalene triangle has all three sides of different lengths.
• It also has all angles unequal.

Last verified on 15-05-2026

Explanation

The decomposition is:

(8x + 6)/[x(x + 2)] = A/x + B/(x + 2)

Multiply both sides by x(x + 2) so the denominators cancel out:

8x + 6 = A(x + 2) + Bx

Now find A and B:

Put x = 0. That makes the Bx term zero, so 6 = 2A. That gives A = 3.  

Put x = -2. That makes the A(x + 2) term zero, so -10 = -2B. That gives B = 5.

So it equals 3/x + 5/(x + 2).

Last verified on 15-05-2026

Explanation

Given y = 18 when x = 3:

18 = k(3²)

18 = 9k

k = 2

Now for x = 5:

y = 2(5²)

y = 2 × 25

y = 50

Last verified on 15-05-2026

Explanation

Find f(3):

f(3) = 3² − 2(3)

= 9 − 6

= 3

Find f(1):

f(1) = 1² − 2(1)

= 1 − 2

= −1

Now add:

f(3) + f(1) = 3 + (−1) = 2

Last verified on 15-05-2026

Explanation

(x - 3)/2 = (x + 1)/4

Multiply both sides by 4 to remove denominators:

4 × (x - 3)/2 = 4 × (x + 1)/4

2(x - 3) = x + 1

Expand:

2x - 6 = x + 1

Bring x terms to one side, numbers to other:

2x - x = 1 + 6

x = 7

Last verified on 15-05-2026

Explanation

Given: x² - 7x + 10 = 0

Step 1: Find original roots

x² - 7x + 10 = (x - 2)(x - 5) = 0

So roots are 2 and 5

Step 2: Multiply roots by 3

New roots = 2 × 3 = 6, 5 × 3 = 15

Step 3: Form new equation

If roots are α and β, equation is x² - (α+β)x + αβ = 0

Here α+β = 6 + 15 = 21

     αβ = 6 × 15 = 90

Therefore new equation = x² - 21x + 90 = 0

Last verified on 15-05-2026

Explanation

Let the two factors be (x - a) and (x - b)

Then:

1. (x - a)(x - b) = x^2 - (a + b)x + ab

2. Compare with x^2 - px + 48:

   ab = 48

   a + b = p

3. Given: a - b = 2 => a = b + 2

Substitute:

(b + 2)b = 48

b^2 + 2b - 48 = 0

(b + 8)(b - 6) = 0

So b = 6 or b = -8

If b = 6, then a = 8, p = a + b = 14

If b = -8, then a = -6, p = a + b = -14

Last verified on 15-05-2026

Explanation

(a + bi)(a − bi) = a² + b²

Given:

(a + bi)(a − bi) = 41

Therefore:

a² + b² = 41

Last verified on 15-05-2026