Solve 3t^2-10t-8=0 by factoring. Enter your answers below t= square t= square

To solve the quadratic equation \(3t^2 - 10t - 8 = 0\) by factoring, we need to find two numbers that multiply to \(3 \times (-8) = -24\) and add up to \(-10\).<br /><br />The two numbers that satisfy these conditions are \(-12\) and \(2\), because:<br />\[<br />-12 \times 2 = -24<br />\]<br />\[<br />-12 + 2 = -10<br />\]<br /><br />Next, we rewrite the middle term \(-10t\) using these two numbers:<br />\[<br />3t^2 - 12t + 2t - 8 = 0<br />\]<br /><br />Now, we factor by grouping:<br />\[<br />3t(t - 4) + 2(t - 4) = 0<br />\]<br /><br />We can factor out the common binomial factor \((t - 4)\):<br />\[<br />(3t + 2)(t - 4) = 0<br />\]<br /><br />Setting each factor equal to zero gives us the solutions:<br />\[<br />3t + 2 = 0 \quad \text{or} \quad t - 4 = 0<br />\]<br /><br />Solving these equations, we get:<br />\[<br />3t + 2 = 0 \implies 3t = -2 \implies t = -\frac{2}{3}<br />\]<br />\[<br />t - 4 = 0 \implies t = 4<br />\]<br /><br />Therefore, the solutions to the equation \(3t^2 - 10t - 8 = 0\) are:<br />\[<br />t = -\frac{2}{3} \quad \text{and} \quad t = 4<br />\]<br /><br />So, the answers are:<br />\[<br />t = -\frac{2}{3} \quad \text{and} \quad t = 4<br />\]