To solve the quadratic equation \( 4x^2 - 9x - 34 = 0 \) using the quadratic formula, we use the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 4 \), \( b = -9 \), and \( c = -34 \).
First, we need to calculate the discriminant, \( b^2 - 4ac \):
\[ b^2 = (-9)^2 = 81 \] \[ 4ac = 4 \cdot 4 \cdot (-34) = -544 \]
Now, calculate the discriminant:
\[ b^2 - 4ac = 81 - (-544) = 81 + 544 = 625 \]
Since the discriminant is positive, we will have two real solutions. Now, we can substitute \( a \), \( b \), and the discriminant into the formula:
\[ x = \frac{-(-9) \pm \sqrt{625}}{2 \cdot 4} \] \[ x = \frac{9 \pm 25}{8} \]
Now, we will calculate the two possible values for \( x \):
-
For the positive case (\( + \)): \[ x_1 = \frac{9 + 25}{8} = \frac{34}{8} = 4.25 \]
-
For the negative case (\( - \)): \[ x_2 = \frac{9 - 25}{8} = \frac{-16}{8} = -2 \]
Thus, the two solutions are \( x_1 = 4.25 \) and \( x_2 = -2 \).
Writing them in decimal form with the lesser value first, we have:
\[ \boxed{-2} \quad \text{and} \quad \boxed{4.25} \]