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Answer:
To solve the quadratic equation x^2 - 6x + 8 = 0 , we can use the quadratic formula or factorization. Let's first try to factor it.
## Step 1: Factor the Quadratic
We want to express the equation in the form (x - p)(x - q) = 0 , where p and q are the roots of the equation. We need two numbers that multiply to 8 (the constant term) and add up to -6 .
The numbers that satisfy these conditions are -2 and -4 because:
- -2 \times -4 = 8
- -2 + (-4) = -6
Thus, we can factor the equation as:
(x - 2)(x - 4) = 0
## Step 2: Set Each Factor to Zero
Now, we set each factor equal to zero to find the solutions:
1. x - 2 = 0 ⟹ x = 2
2. x - 4 = 0 ⟹ x = 4
## Step 3: Solutions
The solutions to the equation x^2 - 6x + 8 = 0 are:
x = 2 \quad \text{and} \quad x = 4
## Verification
To verify, we can substitute these values back into the original equation:
1. For x = 2 :
2^2 - 6(2) + 8 = 4 - 12 + 8 = 0
2. For x = 4 :
4^2 - 6(4) + 8 = 16 - 24 + 8 = 0
Both values satisfy the original equation, confirming that the solutions are correct.