Question

36a²-64c²= 1-a⁴c²= -25c²+9a²=

Answer 1

(a²-b²)=(a-b)(a+b)

36a²-64c²=(6à)²-(8c)²=(6a-8c)(6a+8c)

1-a⁴c²=(1)²-(a²c)²=(1-a²c)(1+a²c)

-25c²+9a²=9a²-25c²=(3a)²-(5c)²=(3a-5c)(3a+5c)

Answer 2

To solve the given system of equations:

1. 36a² - 64c² = 1
2. 1 - a⁴c² = 0
3. -25c² + 9a² = 0

Let's solve them step by step:

From equation (1), we can rewrite it as:

36a² = 64c² + 1

Divide both sides by 36:

a² = (64c² + 1)/36

Now, let's substitute this expression for a² in equations (2) and (3):

2. 1 - [(64c² + 1)/36] * c² = 0

Multiply both sides by 36 to eliminate the denominator:

36 - (64c² + 1) * c² = 0

Expand the expression:

36 - 64c⁴ - c² = 0

Rearrange the terms:

64c⁴ + c² - 36 = 0

3. -25c² + 9[(64c² + 1)/36] = 0

Multiply both sides by 36 to eliminate the denominator:

-25 * 36c² + 9(64c² + 1) = 0

Simplify:

-900c² + 576c² + 9 = 0

Combine like terms:

-324c² + 9 = 0

Now, we have two equations:

1. 64c⁴ + c² - 36 = 0
2. -324c² + 9 = 0

Let's solve equation (2) first:

-324c² + 9 = 0

Adding 324c² to both sides:

9 = 324c²

Divide both sides by 324:

c² = 9/324

Simplify:

c² = 1/36

Taking the square root of both sides:

c = ± 1/6

Now, substitute the values of c into equation (1) to find the corresponding values of a:

For c = 1/6:

36a² = 64(1/6)² + 1 36a² = 64/36 + 1 36a² = 64/36 + 36/36 36a² = 100/36 a² = 100/36 a = ± √(100/36) a = ± (10/6) a = ± 5/3

For c = -1/6:

36a² = 64(-1/6)² + 1 36a² = 64/36 + 1 36a² = 64/36 + 36/36 36a² = 100/36 a² = 100/36 a = ± √(100/36) a = ± (10/6) a = ± 5/3

Therefore, the solutions to the system of equations are:

a = 5/3, c = 1/6 a = -5/3, c = 1/6 a = 5/3, c = -1/6 a = -5/3, c = -1/6