Question

Solve 4x^2−9x−34=0 using the quadratic formula. Write answers in decimal form as needed and give the lesser value first.

Answer 1

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 \( x \):

1. For the positive case (\( + \)):
\[
x_1 = \frac{9 + 25}{8} = \frac{34}{8} = 4.25
\]

2. 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:

\[
\boxed{-2} \quad \text{and} \quad \boxed{4.25}
\]