Questions: The quadratic formula is used to solve for (x) in equations taking the form of a quadratic equation, (a x^2+b x+c=0). quadratic formula: (x=frac-b pm sqrtb^2-4 a c2 a) Solve for (x) in the following expression using the quadratic formula. [2 x^2+29 x-7.5=0] Use at least three significant figures in each answer. [x=] and (x=)

Solution Steps

The given quadratic equation is

\[ 2x^2 + 29x - 7.5 = 0 \]

where \( a = 2 \), \( b = 29 \), and \( c = -7.5 \).

The discriminant \( D \) is calculated using the formula

\[ D = b^2 - 4ac \]

Substituting the values, we have

\[ D = 29^2 - 4 \cdot 2 \cdot (-7.5) = 841 + 60 = 901 \]

Using the quadratic formula

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

we can find the two solutions. Substituting \( D = 901 \), \( a = 2 \), and \( b = 29 \):

\[ x = \frac{-29 \pm \sqrt{901}}{2 \cdot 2} \]

Calculating the two solutions:

1. For the positive root:

\[ x_1 = \frac{-29 + \sqrt{901}}{4} \]

2. For the negative root:

\[ x_2 = \frac{-29 - \sqrt{901}}{4} \]

The solutions to the quadratic equation are:

\[ x = \frac{-29 + \sqrt{901}}{4} \quad \text{and} \quad x = \frac{-29 - \sqrt{901}}{4} \]

These can be approximated as \( x_1 \approx 0.254 \) and \( x_2 \approx -14.754 \) when expressed in decimal form.

Final Answer