Explanation

If base = b, height = h, then area = b × h. 

New base = b × 1.20 = 1.2b  

New height = h × 0.80 = 0.8h  

New area = 1.2b × 0.8h = 0.96bh = 0.96 × original area

Last verified on 24-05-2026

Explanation

Put a = 9 in a² + b² = 145

9² + b² = 145

81 + b² = 145

b² = 145 - 81

b² = 64

b = ±8

Last verified on 24-05-2026

Explanation

• A triangle with all angles equal is an equilateral triangle.
• Each angle in this triangle is 60°.
• All sides are also equal in length.
• It is a regular and perfectly symmetrical triangle.

Last verified on 24-05-2026

Explanation

(10x + 6)/(x(x + 3)) decomposes as 2/x + 8/(x + 3)

Solution:

Let (10x + 6)/(x(x + 3)) = A/x + B/(x + 3)

Multiply both sides by x(x + 3)

10x + 6 = A(x + 3) + B(x)

Put x = 0

10(0) + 6 = A(0 + 3) + B(0)

6 = 3A

A = 2

Put x = -3

10(-3) + 6 = A(-3 + 3) + B(-3)

-30 + 6 = 0 - 3B

-24 = -3B

B = 8

So the decomposition is:

2/x + 8/(x + 3)

Last verified on 24-05-2026

Explanation

If f(x) = x² + 2x then f(2) + f(-2) = 8

Solution:

Put x = 2 in f(x) = x² + 2x

f(2) = 2² + 2_2

f(2) = 4 + 4 = 8

Put x = -2 in f(x) = x² + 2x

f(-2) = (-2)² + 2_(-2)

f(-2) = 4 - 4 = 0

Now add both values

f(2) + f(-2) = 8 + 0 = 8

Last verified on 24-05-2026

Explanation

If y ∝ 1/(x²) and y = 18 when x = 3, then y = 4.5 when x = 6

Solution:

Since y is inversely proportional to x², we can write  

y = k / x², where k is a constant.

When y = 18 and x = 3,

18 = k / 3²

18 = k / 9

k = 18 * 9 = 162

Now put k = 162 and x = 6 to find y,

y = 162 / 6²

y = 162 / 36

y = 4.5

So y = 4.5 when x = 6

Last verified on 24-05-2026

Explanation

Solve: (2x - 1)/3 = (x + 5)/2

Solution:

Step 1: Remove the denominators by cross multiplication

(2x - 1) _ 2 = (x + 5) _ 3

Step 2: Multiply out both sides

4x - 2 = 3x + 15

Step 3: Bring terms with x to left side and constants to right side

4x - 3x = 15 + 2

Step 4: Simplify

x = 17

Last verified on 24-05-2026

Explanation

If roots of x² - 9x + 20 = 0 are increased by 1, the new equation is x² - 11x + 30 = 0

Solution:

Step 1: Find the roots of x² - 9x + 20 = 0

Factorize:

x² - 9x + 20 = 0

x² - 5x - 4x + 20 = 0

x(x - 5) - 4(x - 5) = 0

(x - 4)(x - 5) = 0

So the roots are x = 4 and x = 5

Step 2: Increase each root by 1

New roots = 4 + 1 = 5 and 5 + 1 = 6

Step 3: Form the new equation

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

Here α = 5, β = 6

Sum of roots = 5 + 6 = 11

Product of roots = 5 * 6 = 30

Therefore, new equation is:

x² - 11x + 30 = 0

Last verified on 24-05-2026

Explanation

If x² - px + 72 has factors differing by 1, then p = ±17

Solution in older version:

Let the two factors be m and m+1. 

Then,

x² - px + 72 = (x - m)(x - (m+1))

             = (x - m)(x - m - 1)

             = x² - (2m + 1)x + m(m+1)

Compare with x² - px + 72:

1. m(m+1) = 72

2. p = 2m + 1

Solve m(m+1) = 72:

m² + m - 72 = 0

(m + 9)(m - 8) = 0

So m = 8 or m = -9

Case 1: m = 8

Factors are 8 and 9

p = 2_8 + 1 = 17

Case 2: m = -9  

Factors are -9 and -8

p = 2_(-9) + 1 = -17

Last verified on 24-05-2026

Explanation

(a + bi)(a - bi) is the product of conjugates.  

(a + bi)(a - bi) = a² - (bi)² = a² - b²i²  

Since i² = -1:  

= a² - b²(-1) = a² + b²  

Given (a + bi)(a - bi) = 61,  

so a² + b² = 61.

Last verified on 24-05-2026