Questions: Solve the following polynomial equation by factoring or using the quadratic formula. Identify all solutions. x^2-3 x+28=0

Solution Steps

To solve the quadratic polynomial equation, use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the equation \(ax^2 + bx + c = 0\).

The given quadratic equation is

\[ x^2 - 3x + 28 = 0 \]

From this equation, we identify the coefficients as follows:

• \( a = 1 \)
• \( b = -3 \)
• \( c = 28 \)

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

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

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = (-3)^2 - 4 \cdot 1 \cdot 28 = 9 - 112 = -103 \]

Since the discriminant is negative, the solutions will be complex numbers.

The solutions for \( x \) are given by the quadratic formula:

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

Substituting \( D = -103 \):

\[ x = \frac{-(-3) \pm \sqrt{-103}}{2 \cdot 1} = \frac{3 \pm \sqrt{103}i}{2} \]

This simplifies to:

\[ x_1 = \frac{3}{2} + \frac{\sqrt{103}}{2}i \] \[ x_2 = \frac{3}{2} - \frac{\sqrt{103}}{2}i \]

Final Answer