To solve the equation [tex]\(5(x-8)^2 - 14 = 0\)[/tex], we'll follow these steps:
### Step 1: Isolate the squared term
First, we want to isolate the squared term [tex]\( (x-8)^2 \)[/tex]. To do this, add 14 to both sides of the equation:
[tex]\[ 5(x-8)^2 - 14 + 14 = 0 + 14 \][/tex]
[tex]\[ 5(x-8)^2 = 14 \][/tex]
### Step 2: Divide by the coefficient of the squared term
Next, divide both sides by 5 to solve for [tex]\((x-8)^2\)[/tex]:
[tex]\[ \frac{5(x-8)^2}{5} = \frac{14}{5} \][/tex]
[tex]\[ (x-8)^2 = \frac{14}{5} \][/tex]
[tex]\[ (x-8)^2 = 2.8 \][/tex]
### Step 3: Take the square root of both sides
Now, take the square root of both sides to solve for [tex]\( (x-8) \)[/tex]:
[tex]\[ \sqrt{(x-8)^2} = \pm \sqrt{2.8} \][/tex]
[tex]\[ x - 8 = \pm \sqrt{2.8} \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Add 8 to both sides to find the solutions for [tex]\( x \)[/tex]:
[tex]\[ x = 8 \pm \sqrt{2.8} \][/tex]
### Step 5: Calculate the numerical values
To get the actual numerical values, calculate [tex]\(\sqrt{2.8}\)[/tex]:
[tex]\[ \sqrt{2.8} \approx 1.673 \][/tex]
Then, the solutions for [tex]\( x \)[/tex] are:
[tex]\[ x = 8 + 1.673 \][/tex]
[tex]\[ x \approx 9.673 \][/tex]
And
[tex]\[ x = 8 - 1.673 \][/tex]
[tex]\[ x \approx 6.327 \][/tex]
### Final Step: List the solutions
Therefore, the solutions rounded to three decimal places are:
[tex]\[ x \approx 9.673 \][/tex]
[tex]\[ x \approx 6.327 \][/tex]
These are the solutions to the given equation.
