Explanation

Area of a triangle = 1/2 _ base _ height

Let old base = B, old height = H.  

Old area = 1/2 _ B _ H

New base = B + 25% of B = B + 0.25B = 1.25B  

New height = H - 20% of H = H - 0.20H = 0.80H

New area = 1/2 _ 1.25B _ 0.80H  

= 1/2 _ 1.00 _ B _ H  

= 1/2 _ B * H

New area = old area. It stays the same.

Last verified on 25-05-2026

Explanation

Put a = 1 into a² + b² = 170.

1² + b² = 170  

1 + b² = 170  

b² = 170 - 1  

b² = 169  

b = 13 or b = -13

So b equals 13 or -13.

Last verified on 25-05-2026

Explanation

A triangle is classified based on its angles.

If one angle is 90°, it is a right-angled triangle.

The given triangle has angles 30°, 60°, and 90°.

Since it contains a 90° angle, it is a right triangle.

Last verified on 25-05-2026

Explanation

Write (14x + 10)/(x(x + 5)) = A/x + B/(x + 5)

Multiply both sides by x(x + 5):

14x + 10 = A(x + 5) + Bx

Now find A and B by plugging in values for x:

Put x = 0:

14_0 + 10 = A(0 + 5) + B_0

10 = 5A

A = 2

Put x = -5:

14_(-5) + 10 = A(0) + B_(-5)

-70 + 10 = -5B

-60 = -5B

B = 12

So (14x + 10)/(x(x + 5)) = 2/x + 12/(x + 5)

Last verified on 25-05-2026

Explanation

For f(x) = x² - x, find f(5) and f(3) separately then subtract.

f(5) = 5² - 5 = 25 - 5 = 20  

f(3) = 3² - 3 = 9 - 3 = 6  

Now f(5) - f(3) = 20 - 6 = 14  

Last verified on 25-05-2026

Explanation

y ∝ 1/x³ means y = k / x³ where k is a constant.

Step 1: Find k using y = 32 when x = 2

32 = k / 2³

32 = k / 8

k = 32 * 8 = 256

Step 2: Use k to find y when x = 4

y = 256 / 4³

y = 256 / 64

y = 4

Last verified on 25-05-2026

Explanation

Equation: (4x + 1)/5 = (x + 9)/2

Step 1: Cross multiply to remove fractions

2(4x + 1) = 5(x + 9)

Step 2: Expand both sides

8x + 2 = 5x + 45

Step 3: Move x terms to one side, numbers to the other side

8x - 5x = 45 - 2

3x = 43

Step 4: Divide by 3

x = 43/3

Last verified on 25-05-2026

Explanation

Original equation: x^2 - 11x + 30 = 0

Factor it to find roots:

x^2 - 11x + 30 = (x - 5)(x - 6) = 0

So roots are 5 and 6.

Increase each root by 2:

New roots are 5 + 2 = 7 and 6 + 2 = 8.

New equation with roots 7 and 8 is:

(x - 7)(x - 8) = 0

x^2 - 8x - 7x + 56 = 0

x^2 - 15x + 56 = 0

Last verified on 25-05-2026

Explanation

Let the factors be (x - n) and (x - n - 1).

Multiply: (x - n)(x - n - 1) = x^2 - (2n + 1)x + n(n + 1).

Set n(n + 1) = 110, which gives n = 10.

Then p = 2n + 1 = 21.

Last verified on 25-05-2026

Explanation

If you multiply (a + bi) and (a - bi), you get:

(a + bi)(a - bi) = a_a - a_bi + bi_a - bi_bi

                 = a^2 - abi + abi - b^2_i^2

The middle terms cancel out. And i^2 = -1, so -b^2_i^2 = -b^2*(-1) = b^2.

That leaves a^2 + b^2.

You are told (a + bi)(a - bi) = 89, so a^2 + b^2 = 89.

Last verified on 25-05-2026