Sure, I can help with that. Let's solve these given expressions and fill in the missing terms step-by-step.
1. [tex]\(a^2 - 81 = (a + 9)(a - 9)\)[/tex]
- This is a difference of squares formula: [tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex], where [tex]\(a^2 - 9^2 = (a+9)(a-9)\)[/tex].
2. [tex]\(p^2 - q^2 = (-t)(p - q)\)[/tex]
- Notice that [tex]\(\boldsymbol{p^2 - q^2}\)[/tex] is again a difference of squares, but needs correcting as [tex]\(\boldsymbol{t}\)[/tex] must match the correct formula: [tex]\(\boldsymbol{p^2 - q^2 = (p - q)(p + q)}\)[/tex]
3. [tex]\(c^2 - d^2 = (c + d)(\_\_\_\_\_\_\_\_\_\_)\)[/tex]
- This should be completed with [tex]\((c - d)\)[/tex], based on the difference of squares formula: [tex]\(c^2 - d^2 = (c + d)(c - d)\)[/tex].
Next, let's solve the quadratic equation:
4. Solve [tex]\(25e^2 - 16 = 0\)[/tex]
- Rewrite the equation as [tex]\(25e^2 = 16\)[/tex].
- Divide both sides by 25: [tex]\(e^2 = \frac{16}{25}\)[/tex].
- Take the square root of both sides: [tex]\(e = \pm \frac{4}{5}\)[/tex].
So the solutions for [tex]\(e\)[/tex] are:
[tex]\[
e_{positive} = \frac{4}{5} = 0.8 \quad and \quad e_{negative} = -\frac{4}{5} = -0.8
\][/tex]
For the expression involving [tex]\(r\)[/tex] and [tex]\(l\)[/tex]:
5. Solve for [tex]\(l\)[/tex] in terms of [tex]\(r\)[/tex]:
- [tex]\(r^2 + 95^4 = l\)[/tex].
Let's use a specific example value for [tex]\(r = 1\)[/tex]:
[tex]\[
l = r^2 + 95^4 = 1^2 + 95^4 = 1 + 81450625 = 81450626
\][/tex]
To summarize:
1. [tex]\(a^2 - 81 = (a + 9)(a - 9)\)[/tex]
2. [tex]\(p^2 - q^2 = (p + q)(p - q) \)[/tex]
3. [tex]\(c^2 - d^2 = (c + d)(c - d)\)[/tex]
4. Solutions for [tex]\(25e^2 - 16 = 0\)[/tex] are:
[tex]\[
e = 0.8 \quad \text{and} \quad e = -0.8
\][/tex]
5. With [tex]\(r = 1\)[/tex]:
[tex]\[
r^2 + 95^4 = l \Rightarrow l = 81450626
\][/tex]
