Let's go through the problem step-by-step to solve the given quadratic equation.
### Step 1: Put the equation in standard form
The given equation is:
[tex]\[ 11x^2 - 3x - 6 = 4 \][/tex]
First, we need to move all terms to one side to get the equation in its standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Subtract 4 from both sides:
[tex]\[ 11x^2 - 3x - 6 - 4 = 0 \][/tex]
This simplifies to:
[tex]\[ 11x^2 - 3x - 10 = 0 \][/tex]
### Step 2: Identify coefficients
Now that we have the quadratic equation in standard form [tex]\( 11x^2 - 3x - 10 = 0 \)[/tex], we can identify the coefficients:
- [tex]\( a = 11 \)[/tex]
- [tex]\( b = -3 \)[/tex]
- [tex]\( c = -10 \)[/tex]
### Step 3: Calculate the discriminant
The discriminant ([tex]\( \Delta \)[/tex]) of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[ \Delta = (-3)^2 - 4(11)(-10) \][/tex]
[tex]\[ \Delta = 9 + 440 \][/tex]
[tex]\[ \Delta = 449 \][/tex]
### Step 4: Calculate the solutions using the quadratic formula
The quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex] provides the solutions for the equation. Now, let's substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex] into the formula:
[tex]\[ x = \frac{-(-3) \pm \sqrt{449}}{2 \times 11} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{449}}{22} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{3 + \sqrt{449}}{22} \][/tex]
[tex]\[ x_2 = \frac{3 - \sqrt{449}}{22} \][/tex]
### Step 5: Compute the actual values of [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]
For a more accurate result, we calculate the numerical values:
[tex]\[ \sqrt{449} \approx 21.1896 \][/tex]
Therefore,
[tex]\[ x_1 = \frac{3 + 21.1896}{22} \approx \frac{24.1896}{22} \approx 1.0995 \][/tex]
[tex]\[ x_2 = \frac{3 - 21.1896}{22} \approx \frac{-18.1896}{22} \approx -0.8268 \][/tex]
### Solution
The solutions to the quadratic equation [tex]\( 11x^2 - 3x - 10 = 0 \)[/tex] are approximately:
[tex]\[ x_1 \approx 1.0995 \][/tex]
[tex]\[ x_2 \approx -0.8268 \][/tex]
In summary:
- Discriminant (Δ): 449
- Roots: [tex]\( x_1 \approx 1.0995 \)[/tex] and [tex]\( x_2 \approx -0.8268 \)[/tex]
