Questions: Find the discriminant to determine the number of roots, then solve the quadratic equation using the quadratic formula. x^2+x-2=0 discriminant number of roots solution a. 0 b. -1 and 2 c. -9 d. 9 e. 2 f. 1 g. 1 and -2

Solution Steps

To solve the quadratic equation \(x^2 + x - 2 = 0\), we need to:

1. Calculate the discriminant using the formula \(b^2 - 4ac\).
2. Determine the number of roots based on the discriminant.
3. Solve the quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

The quadratic equation is given by: \[ x^2 + x - 2 = 0 \] The discriminant \(\Delta\) is calculated using the formula: \[ \Delta = b^2 - 4ac \] Substituting \(a = 1\), \(b = 1\), and \(c = -2\): \[ \Delta = 1^2 - 4 \cdot 1 \cdot (-2) = 1 + 8 = 9 \]

The number of roots of the quadratic equation depends on the value of the discriminant \(\Delta\):

• If \(\Delta > 0\), there are 2 distinct real roots.
• If \(\Delta = 0\), there is 1 real root.
• If \(\Delta < 0\), there are no real roots.

Since \(\Delta = 9 > 0\), there are 2 distinct real roots.

The roots of the quadratic equation are given by the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting \(a = 1\), \(b = 1\), and \(\Delta = 9\): \[ x = \frac{-1 \pm \sqrt{9}}{2 \cdot 1} = \frac{-1 \pm 3}{2} \] This gives us two solutions: \[ x_1 = \frac{-1 + 3}{2} = \frac{2}{2} = 1 \] \[ x_2 = \frac{-1 - 3}{2} = \frac{-4}{2} = -2 \]

Final Answer