Let's solve the quadratic equation [tex]\(2(m - x)^2 = 18\)[/tex] for [tex]\(x\)[/tex], where [tex]\(m\)[/tex] is a constant.
Step 1: Isolate the quadratic term
First, let's divide both sides of the equation by 2 to simplify:
[tex]\[ \frac{2(m - x)^2}{2} = \frac{18}{2} \][/tex]
[tex]\[ (m - x)^2 = 9 \][/tex]
Step 2: Take the square root of both sides
Next, take the square root of both sides to solve for the absolute value:
[tex]\[ \sqrt{(m - x)^2} = \sqrt{9} \][/tex]
[tex]\[ |m - x| = 3 \][/tex]
Step 3: Solve the absolute value equation
The equation [tex]\( |m - x| = 3 \)[/tex] means that [tex]\( m - x \)[/tex] could be either 3 or -3. Therefore, we have two cases to consider:
Case 1: [tex]\( m - x = 3 \)[/tex]
[tex]\[ m - x = 3 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = m - 3 \][/tex]
Case 2: [tex]\( m - x = -3 \)[/tex]
[tex]\[ m - x = -3 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = m + 3 \][/tex]
Conclusion
The solutions for the quadratic equation [tex]\( 2(m - x)^2 = 18 \)[/tex] are:
[tex]\[ x = m - 3 \][/tex]
[tex]\[ x = m + 3 \][/tex]
So, the possible values of [tex]\( x \)[/tex] are [tex]\( m - 3 \)[/tex] and [tex]\( m + 3 \)[/tex].
