Questions: The solutions to the equation (ax^2+bx+c=0) are given by the quadratic formula. The formula contains a (pm) symbol, but it is important to recognize that it is actually producing two different answers (usually). Write separately the solutions given by the quadratic formula in terms of (a, b), and (c) below. First the solution with the + sign: (x=) Next, the solution with the - sign: (x=)

Solution Steps

To solve the quadratic equation \( ax^2 + bx + c = 0 \) using the quadratic formula, we need to calculate the two possible solutions. The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula provides two solutions: one using the plus sign and the other using the minus sign. We will compute these two solutions separately.

1. Calculate the discriminant, \( D = b^2 - 4ac \).
2. Compute the first solution using the plus sign: \( x_1 = \frac{-b + \sqrt{D}}{2a} \).
3. Compute the second solution using the minus sign: \( x_2 = \frac{-b - \sqrt{D}}{2a} \).

For the quadratic equation \( ax^2 + bx + c = 0 \) with coefficients \( a = 1 \), \( b = -3 \), and \( c = 2 \), we first calculate the discriminant \( D \):

\[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 2 = 9 - 8 = 1 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \), we find the two solutions.

Solution with the + sign: \[ x_1 = \frac{-(-3) + \sqrt{1}}{2 \cdot 1} = \frac{3 + 1}{2} = \frac{4}{2} = 2.0 \]

Solution with the - sign: \[ x_2 = \frac{-(-3) - \sqrt{1}}{2 \cdot 1} = \frac{3 - 1}{2} = \frac{2}{2} = 1.0 \]

Final Answer