Questions: Solve the equation after making an appropriate substitution. Complete (19 x-2)^2-2(19 x-2)-8=0 a) Determine the appropriate substitution using the new variable u. u =

Solution Steps

Let \( u = 19x - 2 \). This substitution simplifies the equation by replacing the expression \( 19x - 2 \) with \( u \).

Substitute \( u = 19x - 2 \) into the original equation: \[ u^2 - 2u - 8 = 0 \]

Solve the quadratic equation \( u^2 - 2u - 8 = 0 \) using the quadratic formula: \[ u = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \] \[ u = \frac{2 \pm \sqrt{4 + 32}}{2} \] \[ u = \frac{2 \pm \sqrt{36}}{2} \] \[ u = \frac{2 \pm 6}{2} \] This gives two solutions: \[ u = \frac{2 + 6}{2} = 4 \quad \text{and} \quad u = \frac{2 - 6}{2} = -2 \]

Now, substitute \( u = 19x - 2 \) back into the solutions for \( u \):

1. For \( u = 4 \): \[ 19x - 2 = 4 \] \[ 19x = 6 \] \[ x = \frac{6}{19} \]

2. For \( u = -2 \): \[ 19x - 2 = -2 \] \[ 19x = 0 \] \[ x = 0 \]

Final Answer