Explanation

Given a + b + c = 9 and ab + bc + ca = 20

We know that (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values:

(9)² = a² + b² + c² + 2(20)

81 = a² + b² + c² + 40

Now, find a² + b² + c²:

a² + b² + c² = 81 - 40

= 41

Explanation

Given logx (4/9) = 2

This can be rewritten as:

x² = 4/9

Taking the square root of both sides:

x = ±√(4/9)

x = ±2/3

Since the base of a logarithm is typically positive, we consider x = 2/3.

Explanation

The given quadratic expression is x² - 2x + 1.

This can be factored as:

x² - 2x + 1 = (x - 1)(x - 1)

= (x - 1)²

The answer is (x - 1)(x - 1).

Explanation

Let's solve the equation:

4/5 (x + 7) = 3(2x - 7)

Multiply both sides by 5 to eliminate the fraction:

4(x + 7) = 15(2x - 7)

Expand both sides:

4x + 28 = 30x - 105

Subtract 4x from both sides:

28 = 26x - 105

Add 105 to both sides:

133 = 26x

Divide both sides by 26:

x = 133/26

Explanation

Given cos θ = 4/5 and θ lies in the 4th quadrant.

In the 4th quadrant, cosine is positive, and sine is negative.

We know that sin² θ + cos² θ = 1

sin² θ = 1 - cos² θ

= 1 - (4/5)²

= 1 - 16/25

= 9/25

sin θ = ±√(9/25)

= ±3/5

Since θ lies in the 4th quadrant, sin θ is negative:

sin θ = -3/5

Explanation

Given that x + 1 is a factor of the polynomial x³ + ax² - x + 2, we know that (-1) is a root of the polynomial.

Let's plug in x = -1 into the polynomial:

(-1)³ + a(-1)² - (-1) + 2 = 0

-1 + a + 1 + 2 = 0

a + 2 = 0

Subtract 2 from both sides:

a = -2

Explanation

Given:

a² + b² + c² = 12

a + b + c = 5

We know that:

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Expanding the equation:

5² = 12 + 2(ab + bc + ca)

25 = 12 + 2(ab + bc + ca)

Subtract 12 from both sides:

13 = 2(ab + bc + ca)

Divide by 2:

ab + bc + ca = 13/2

Explanation

Given:

logx 0.25 = 2

This can be rewritten in exponential form:

x² = 0.25

Taking the square root of both sides:

x = √0.25

x = 0.5

Explanation

To factorize the quadratic expression x² - 12x + 35, we need to find two numbers whose product is 35 and whose sum is -12.

The numbers are -7 and -5, since (-7) * (-5) = 35 and (-7) + (-5) = -12.

So, the factorization is:

x² - 12x + 35 = (x - 7)(x - 5)

Explanation

2x=32

Step 1: Express 32 as a power of 2

32=2^5

Step 2: Set the exponents equal

2^x=2^5⇒x=5

 Correct answer: 5