Question

Solve for t.
3(t²-1) = 2t²+4t+1​

Answer 1

Exact Form:

t = 2 ± 2√2

Decimal Form:

t = 4.82842712, − 0.82842712

Explanation:

Answer 2

`\bf 3(t^2-1)=2t^2+4t+1\implies 3t^2-3=2t^2+4t+1\implies t^2-3=4t+1 \\\\\\ t^2-4t-3=1\implies t^2-4t-4=0 \\\\\\ ~~~~~~~~~~~~\textit{quadratic formula} \\\\ \stackrel{\stackrel{a}{\downarrow }}{1}t^2\stackrel{\stackrel{b}{\downarrow }}{-4}t\stackrel{\stackrel{c}{\downarrow }}{-4} \qquad \qquad t= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ t=\cfrac{-(-4)\pm√((-4)^2-4(1)(-4))}{2(1)}\implies t=\cfrac{4\pm√(16+16)}{2}`

`\bf t=\cfrac{4\pm√(4^2+4^2)}{2}\implies t=\cfrac{4\pm√(2(4^2))}{2}\implies t=\cfrac{\stackrel{2}{~~\begin{matrix} 4 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\pm \stackrel{2}{~~\begin{matrix} 4 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}√(2)}{~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~} \\\\\\ t = 2\pm 2√(2)\implies t\approx \begin{cases} 4.828427124746191\\ -0.8284271247461907 \end{cases}`