Solve algebraically
x^2-6x+9>0
<=>(x-3)^2>0 <=> -infinity<x<infinity
i applied a^2-2ab+b^2=(a-b)^2
x^2 - 6x + 9 > 0.
(x-3)^2 > 0,
Take sqrt of both sides:
x-3 > 0, X > 3.
To solve the inequality algebraically, we’ll follow these steps:
Step 1: Factor the quadratic equation
Step 2: Find the x-intercepts
Step 3: Determine the sign of the quadratic equation in each interval
Step 4: Write the solution
Step 1: Factor the quadratic equation
To factor x^2 - 6x + 9, we need to find two numbers whose product is 9 and whose sum is -6. The numbers are -3 and -3, because (-3)*(-3) = 9 and (-3) + (-3) = -6. Therefore, we can rewrite the equation as:
(x - 3)(x - 3) > 0
Step 2: Find the x-intercepts
To find the x-intercepts, we equate each factor to zero:
x - 3 = 0
x = 3
Step 3: Determine the sign of the quadratic equation in each interval
Now, we need to determine the sign of the expression (x - 3)(x - 3) over different intervals. We can choose any value less than 3, between 3 and infinity, and greater than 3. Let’s consider the values -4, 0, and 4.
For x = -4:
(-4 - 3)(-4 - 3) = (-7)(-7) = 49 > 0
For x = 0:
(0 - 3)(0 - 3) = (-3)(-3) = 9 > 0
For x = 4:
(4 - 3)(4 - 3) = (1)(1) = 1 > 0
From this, we can see that the expression (x - 3)(x - 3) is positive (+) for all values of x less than 3 and for all values of x greater than 3.
Step 4: Write the solution
To write the solution, we use interval notation:
(-∞, 3) ∪ (3, ∞)
Therefore, the solution to the inequality x^2 - 6x + 9 > 0 is (-∞, 3) ∪ (3, ∞).