a-b=5 and ab=36 then what is the value of a^3 - b^3

asked by guest
on Jan 19, 2025 at 1:18 am



You asked:

Given the following equations: \[\begin{aligned}a - b &= 5\\ a b &= 36\end{aligned}\] Evaluate \({a}^{3} - {b}^{3}\).

MathBot Answer:

The system of equations has \(2\) solutions.

\[a = -4, b = -9\]\[a = 9, b = 4\]

By substituting these values into the expression, we find that: $$\begin{aligned}a^{3} - b^{3} &= 665\end{aligned}$$

Solve \(a - b = 5\) for \(a\). \[a = b + 5\]Substitute \(b + 5\) for \(a\) in \(a b = 36\) and simplify. $$\begin{aligned}a b &= 36 \\ \left(b + 5\right) b &= 36 \\ b^{2} + 5 b &= 36 \\b^{2} + 5 b - 36 &= 0 \\ \left(b - 4\right) \left(b + 9\right) &= 0 \\ b = -9&, b = 4\end{aligned}$$Substitute \(-9\) into \(a - b = 5\) to solve for \(a\). \[\begin{aligned}a + 9 &= 5\\a &= -4\end{aligned}\]This yields the following solution. $$\begin{aligned}a = -4,\,b = -9\end{aligned}$$Substitute \(4\) into \(a - b = 5\) to solve for \(a\). \[\begin{aligned}a - 4 &= 5\\a &= 9\end{aligned}\]This yields the following solution. $$\begin{aligned}a = 9,\,b = 4\end{aligned}$$

By substituting these values into the expression, we find that: $$\begin{aligned}a^{3} - b^{3} &= 665\end{aligned}$$

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