Alright, let's dive into this problem step-by-step.
Given that [tex]\( x = \frac{1}{4} \)[/tex]:
### Part (a): [tex]\( x(x+5) \)[/tex]
1. Substitute [tex]\( x = \frac{1}{4} \)[/tex] into the expression:
[tex]\[
\left( \frac{1}{4} \right) \left( \frac{1}{4} + 5 \right)
\][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[
\frac{1}{4} + 5 = \frac{1}{4} + \frac{20}{4} = \frac{21}{4}
\][/tex]
3. Now, multiply [tex]\( \frac{1}{4} \)[/tex] by [tex]\( \frac{21}{4} \)[/tex]:
[tex]\[
\frac{1}{4} \times \frac{21}{4} = \frac{21}{16}
\][/tex]
Therefore, the solution for part (a) is:
[tex]\[
\frac{21}{16}
\][/tex]
### Part (b): [tex]\( (x+3)(x-2) \)[/tex]
1. Substitute [tex]\( x = \frac{1}{4} \)[/tex] into the expression:
[tex]\[
\left( \frac{1}{4} + 3 \right) \left( \frac{1}{4} - 2 \right)
\][/tex]
2. Simplify both expressions inside the parentheses:
[tex]\[
\frac{1}{4} + 3 = \frac{1}{4} + \frac{12}{4} = \frac{13}{4}
\][/tex]
[tex]\[
\frac{1}{4} - 2 = \frac{1}{4} - \frac{8}{4} = \frac{-7}{4}
\][/tex]
3. Now, multiply [tex]\( \frac{13}{4} \)[/tex] by [tex]\( \frac{-7}{4} \)[/tex]:
[tex]\[
\frac{13}{4} \times \frac{-7}{4} = \frac{-91}{16}
\][/tex]
Therefore, the solution for part (b) is:
[tex]\[
\frac{-91}{16}
\][/tex]
To summarize:
- Part (a) results in [tex]\( \frac{21}{16} \)[/tex].
- Part (b) results in [tex]\( \frac{-91}{16} \)[/tex].
