Questions: A 10 x^2+9 x-9 A=10 x B=9 x C=-9

Solution Steps

The given expression is a quadratic equation in the form \( ax^2 + bx + c \). To solve this, we can use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 10 \), \( b = 9 \), and \( c = -9 \). We will calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots and then use the quadratic formula to find the solutions.

The discriminant \( D \) is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values \( a = 10 \), \( b = 9 \), and \( c = -9 \): \[ D = 9^2 - 4 \cdot 10 \cdot (-9) = 81 + 360 = 441 \]

Since the discriminant \( D = 441 \) is positive, this indicates that there are two distinct real roots.

The roots can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values: \[ x = \frac{-9 \pm \sqrt{441}}{2 \cdot 10} \] Calculating \( \sqrt{441} = 21 \): \[ x = \frac{-9 \pm 21}{20} \]

Calculating the two possible values for \( x \):

1. For the positive root: \[ x_1 = \frac{-9 + 21}{20} = \frac{12}{20} = \frac{3}{5} \]
2. For the negative root: \[ x_2 = \frac{-9 - 21}{20} = \frac{-30}{20} = -\frac{3}{2} \]

Final Answer