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Solve. \[ (5 t+3)(2 t-6)=0 \] (Simplify your answer. Use a comma to separate answers as needed. Express complex numbers in terms of \( i \). Type each solution only once.)

Ask by Washington Bowman. in the United States
Feb 20,2025

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\( t = -\frac{3}{5}, 3 \)

Solución

Solve the equation \( (5t+3)(2t-6)=0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(\left(5t+3\right)\left(2t-6\right)=0\) - step1: Separate into possible cases: \(\begin{align}&2t-6=0\\&5t+3=0\end{align}\) - step2: Solve the equation: \(\begin{align}&t=3\\&t=-\frac{3}{5}\end{align}\) - step3: Rewrite: \(t_{1}=-\frac{3}{5},t_{2}=3\) The solutions to the equation \((5t+3)(2t-6)=0\) are \(t=-\frac{3}{5}\) and \(t=3\).

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Bonus Knowledge

To solve the equation \( (5t + 3)(2t - 6) = 0 \), we can apply the zero product property, which states that if the product of two factors equals zero, then at least one of the factors must be zero. First, set each factor equal to zero: 1. \( 5t + 3 = 0 \) Subtract 3 from both sides: \[ 5t = -3 \] Now, divide by 5: \[ t = -\frac{3}{5} \] 2. \( 2t - 6 = 0 \) Add 6 to both sides: \[ 2t = 6 \] Now, divide by 2: \[ t = 3 \] Thus, the solutions are: \[ -\frac{3}{5}, 3 \]

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