Questions: UNIT 4 - CHALLENGE 4.3: Graphing Quadratic 5 - Quadratic Equations with No Real Solution LEARNING OBJECTIVE: Determine if a quadratic equation has real or nonreal solutions by finding the value of the discriminant. For the quadratic equation (2 x^2+x-6=0), use the discriminant to determine if the solutions to the equation are real or non-real. a.) Discriminant is positive; non-real solutions b.) Discriminant is negative; non-real solutions c.) Discriminant is positive; real solutions d.) Discriminant is negative; real solutions

Solution Steps

For the quadratic equation \(2x^2 + x - 6 = 0\), identify the coefficients \(a\), \(b\), and \(c\):

• \(a = 2\)
• \(b = 1\)
• \(c = -6\)

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by: \[ \Delta = b^2 - 4ac \] Substitute the values of \(a\), \(b\), and \(c\) into the formula: \[ \Delta = 1^2 - 4(2)(-6) = 1 + 48 = 49 \]

• If \(\Delta > 0\), the quadratic equation has two distinct real solutions.
• If \(\Delta = 0\), the quadratic equation has exactly one real solution.
• If \(\Delta < 0\), the quadratic equation has two non-real (complex) solutions.

Since \(\Delta = 49\) and \(49 > 0\), the quadratic equation has two distinct real solutions.

Final Answer