a+b=8

a*b=15

a**2+b**2=?

asked by guest
on Nov 29, 2024 at 6:39 am



You asked:

Given the following equations: \[\begin{aligned}a + b &= 8\\ a b &= 15\end{aligned}\] Evaluate \({a}^{2} + {b}^{2}\).

MathBot Answer:

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

\[a = 3, b = 5\]\[a = 5, b = 3\]

By substituting these values into the expression, we find that: $$\begin{aligned}a^{2} + b^{2} &= 34\end{aligned}$$

Solve \(a + b = 8\) for \(a\). \[a = 8 - b\]Substitute \(8 - b\) for \(a\) in \(a b = 15\) and simplify. $$\begin{aligned}a b &amp= 15 \\ \left(8 - b\right) b &= 15 \\ b^{2} - 8 b &= -15 \\b^{2} - 8 b + 15 &= 0 \\ \left(b - 5\right) \left(b - 3\right) &= 0 \\ b = 3&, b = 5\end{aligned}$$Substitute \(3\) into \(a + b = 8\) to solve for \(a\). \[\begin{aligned}a + 3 &= 8\\a &= 5\end{aligned}\]This yields the following solution. $$\begin{aligned}a = 5,\,b = 3\end{aligned}$$Substitute \(5\) into \(a + b = 8\) to solve for \(a\). \[\begin{aligned}a + 5 &= 8\\a &= 3\end{aligned}\]This yields the following solution. $$\begin{aligned}a = 3,\,b = 5\end{aligned}$$

By substituting these values into the expression, we find that: $$\begin{aligned}a^{2} + b^{2} &= 34\end{aligned}$$

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