Question
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Answer
Tenglama \(4^{x^{2} - 3x - 4} = 1\) yechimlari \(x = 4\) va \(x = -1\).
Solution
e) \(4^{x^{2} - 3x - 4} = 1\)
Bu tenglamani yechish uchun quyidagi qadamlarni bajaramiz:
1. **Asosiy tushuncha:**
\[
4^{y} = 1
\]
Tenglama \(4^{y} = 1\) bo'lganda, bu faqat \(y = 0\) uchun to'g'ri bo'ladi, chunki \(4^{0} = 1\).
2. **Exponentni nolga tenglash:**
Shu sababli, exponentni nolga tenglaymiz:
\[
x^{2} - 3x - 4 = 0
\]
3. **Kvadrat tenglamani yechish:**
Kvadrat tenglama \(ax^{2} + bx + c = 0\) shaklida bo'lib, uning yechimini topish uchun kvadrat formuladan foydalanamiz:
\[
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
\]
Bizning tenglamamizda \(a = 1\), \(b = -3\), va \(c = -4\). Shu qiymatlarni formulaga qo'yamiz:
\[
x = \frac{3 \pm \sqrt{(-3)^{2} - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2}
\]
4. **Yeсimlarni topish:**
\[
x = \frac{3 + 5}{2} = \frac{8}{2} = 4
\]
\[
x = \frac{3 - 5}{2} = \frac{-2}{2} = -1
\]
**Natija:** Tenglama uchun yechimlar \(x = 4\) va \(x = -1\).
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Simplify this solution Beyond the Answer
To solve \( 4^{x^{2}-3 x-4}=1 \), we begin by recognizing that any number (except zero) raised to the power of zero equals one. This implies \( x^2 - 3x - 4 = 0 \). Next, we can factor the quadratic equation as follows: \[ x^2 - 3x - 4 = (x - 4)(x + 1) = 0. \] Setting each factor to zero gives us the solutions \( x - 4 = 0 \) or \( x + 1 = 0 \), leading to \( x = 4 \) and \( x = -1 \). Thus, the final solutions are \( x = 4 \) and \( x = -1 \).
