For a quadratic equation in the form of ax² + bx + c = 0, the sum of the roots is given by -b/a.
Given the equation px² - qx + r = 0:
Step 1: Identify a and b
a = p, b = -q
Step 2: Calculate the sum of roots
Sum of roots = -(-q) / p
= q / p
Let's evaluate the expression step by step:
1. Calculate 2022²:
2022² = 4,088,484
2. Calculate 1²:
1² = 1
3. Calculate 2022² - 1²:
4,088,484 - 1 = 4,088,483
4. Divide by 2021:
4,088,483 ÷ 2021 = 2023
Given the equation and solution path:
5^x + 2 - 5x = 600
If we interpret it as 5^(x+2) - 5^x = 600 (based on your calculation):
5^(2+2) - 5^2 = 5^4 - 5^2 = 625 - 25 = 600
This calculation implies x = 2 fits perfectly if the equation is treated as 5^(x+2) - 5^x = 600.
The value of x is 2.
In the Cartesian coordinate system:
Given P(2, -3), the x-coordinate is positive (+) and the y-coordinate is negative (-), so it lies in the 4th quadrant.
Given the expression a² - 2a + 1 - b²:
Step 1: Factor the perfect square trinomial
a² - 2a + 1 = (a - 1)²
So, the expression becomes (a - 1)² - b²
Step 2: Apply the difference of squares formula
(a - 1)² - b² = (a - 1 - b)(a - 1 + b)
Given y ∝ x, we can write y = kx.
When x = 5, y = 210:
210 = k(5)
k = 210/5
k = 42
Now, when x = 2:
y = kx
y = 42(2)
y = 84
= √(200 + 6 × 10 - 4)/ 16
= √(200 + 60 - 4)/ 16
= √(260 - 4)/ 16
= √(256)/ 16
= 16 / 16
= 1
4^(4x+1) = 1/64
We can start by noticing that 1/64 = 2^(-6) = 4^(-3) (since 4 = 2^2).
So, we have:
4^(4x+1) = 4^(-3)
Since the bases are the same (both are 4), we can equate the exponents:
4x + 1 = -3
Subtracting 1 from both sides gives:
4x = -4
Dividing both sides by -4 gives:
x = -1
Subtracting a negative number is the same as adding its positive:
(-16) - (-4) = -16 + 4
-16 + 4 = -12
Given A and B are square matrices of the same order:
(AB)^t = B^t * A^t
This property holds true for matrix transpose in multiplication.
The correct answer is (AB)^t = B^t * A^t.