Let's solve the equation step by step.
### Step 1: Put the equation in standard form
The given equation is:
[tex]\[ 11x^2 - 3x - 6 = 4 \][/tex]
First, we need to get the equation in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. To do this, subtract 4 from both sides of the equation:
[tex]\[ 11x^2 - 3x - 6 - 4 = 0 \][/tex]
[tex]\[ 11x^2 - 3x - 10 = 0 \][/tex]
Now the equation is in standard form where:
- [tex]\( a = 11 \)[/tex]
- [tex]\( b = -3 \)[/tex]
- [tex]\( c = -10 \)[/tex]
### Step 2: Use the quadratic formula to solve for [tex]\( x \)[/tex]
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
1. Calculate the Discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = (-3)^2 - 4 \cdot 11 \cdot (-10) \][/tex]
[tex]\[ \text{Discriminant} = 9 + 440 \][/tex]
[tex]\[ \text{Discriminant} = 449 \][/tex]
2. Calculate the square root of the Discriminant:
[tex]\[ \sqrt{\text{Discriminant}} = \sqrt{449} \][/tex]
[tex]\[ \sqrt{449} \approx 21.18962010041709 \][/tex]
3. Calculate the roots using the quadratic formula:
[tex]\[ x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a} \][/tex]
[tex]\[ x_1 = \frac{-(-3) + 21.18962010041709}{2 \cdot 11} \][/tex]
[tex]\[ x_1 = \frac{3 + 21.18962010041709}{22} \][/tex]
[tex]\[ x_1 \approx \frac{24.18962010041709}{22} \][/tex]
[tex]\[ x_1 \approx 1.099528186382595 \][/tex]
[tex]\[ x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a} \][/tex]
[tex]\[ x_2 = \frac{-(-3) - 21.18962010041709}{2 \cdot 11} \][/tex]
[tex]\[ x_2 = \frac{3 - 21.18962010041709}{22} \][/tex]
[tex]\[ x_2 \approx \frac{-18.18962010041709}{22} \][/tex]
[tex]\[ x_2 \approx -0.8268009136553224 \][/tex]
### Final Solution:
The roots of the equation [tex]\( 11x^2 - 3x - 10 = 0 \)[/tex] are:
[tex]\[ x_1 \approx 1.099528186382595 \][/tex]
[tex]\[ x_2 \approx -0.8268009136553224 \][/tex]
