23. \( 4(t-1)^{2}-9(t-1)=-2 \)

To solve the equation \( 4(t-1)^{2}-9(t-1)=-2 \), first, let's simplify it by substituting \( x = t - 1 \). This transforms the equation into \( 4x^2 - 9x + 2 = 0 \). Now you can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) using \( a = 4 \), \( b = -9 \), and \( c = 2 \). Now, plug those values into the quadratic formula: \( x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 4 \cdot 2}}{2 \cdot 4} \). Calculate the discriminant \( 81 - 32 = 49 \), which simplifies to \( x = \frac{9 \pm 7}{8} \). This gives you two possible values of \( x \): \( x = 2 \) and \( x = \frac{1}{4} \). Since we substituted \( x = t - 1 \), convert back to find \( t \): 1. For \( x = 2 \): \( t - 1 = 2 \) leads to \( t = 3 \). 2. For \( x = \frac{1}{4} \): \( t - 1 = \frac{1}{4} \) leads to \( t = \frac{5}{4} \). Thus, the solutions are \( t = 3 \) and \( t = \frac{5}{4} \).