What are the solutions to the equation (x-2)(x-4)= 8?
The Correct Answer and Explanation is:
To solve the equation ((x – 2)(x – 4) = 8), we will begin by expanding the left-hand side and then solving the resulting quadratic equation.
Step 1: Expand the Left-Hand Side
First, apply the distributive property (also known as the FOIL method for binomials) to expand ((x – 2)(x – 4)).
[
(x – 2)(x – 4) = x(x – 4) – 2(x – 4)
]
[
= x^2 – 4x – 2x + 8
]
[
= x^2 – 6x + 8
]
So, the equation becomes:
[
x^2 – 6x + 8 = 8
]
Step 2: Subtract 8 from Both Sides
Now, subtract 8 from both sides of the equation to set it equal to 0:
[
x^2 – 6x + 8 – 8 = 8 – 8
]
[
x^2 – 6x = 0
]
Step 3: Factor the Quadratic Expression
Next, factor the quadratic equation (x^2 – 6x = 0). We can factor out (x) from both terms:
[
x(x – 6) = 0
]
Step 4: Solve for (x)
Now, set each factor equal to 0:
[
x = 0 \quad \text{or} \quad x – 6 = 0
]
[
x = 0 \quad \text{or} \quad x = 6
]
Step 5: Verify the Solutions
Finally, we can verify the solutions by substituting (x = 0) and (x = 6) into the original equation ((x – 2)(x – 4) = 8):
- For (x = 0):
[
(0 – 2)(0 – 4) = (-2)(-4) = 8 \quad \text{(True)}
] - For (x = 6):
[
(6 – 2)(6 – 4) = (4)(2) = 8 \quad \text{(True)}
]
Final Answer:
Thus, the solutions to the equation ((x – 2)(x – 4) = 8) are:
[
x = 0 \quad \text{or} \quad x = 6
]