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Sagot :
Alright, let's solve these two equations step by step.
### 1. Solving the quadratic equation [tex]\( x^2 + 6x + 11 = 0 \)[/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
#### Step 1: Identify coefficients
Here,
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 6 \][/tex]
[tex]\[ c = 11 \][/tex]
#### Step 2: Calculate the discriminant
The discriminant [tex]\( \Delta \)[/tex] is calculated using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values:
[tex]\[ \Delta = 6^2 - 4(1)(11) \][/tex]
[tex]\[ \Delta = 36 - 44 \][/tex]
[tex]\[ \Delta = -8 \][/tex]
Since the discriminant is negative, the equation has two complex roots.
#### Step 3: Calculate the complex roots
The roots are given by the formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since [tex]\( \Delta \)[/tex] is negative, we can write it as:
[tex]\[ \sqrt{\Delta} = \sqrt{-8} = 2i\sqrt{2} \][/tex]
Now, substituting the values:
[tex]\[ x = \frac{-6 \pm 2i\sqrt{2}}{2 \times 1} \][/tex]
[tex]\[ x = \frac{-6 \pm 2i\sqrt{2}}{2} \][/tex]
[tex]\[ x = -3 \pm i\sqrt{2} \][/tex]
So, the solutions are:
[tex]\[ x_1 = -3 + i\sqrt{2} \][/tex]
[tex]\[ x_2 = -3 - i\sqrt{2} \][/tex]
### 2. Solving [tex]\( 2x^2 - 16x = 50 \)[/tex] by completing the square
#### Step 1: Move the constant to the other side
[tex]\[ 2x^2 - 16x - 50 = 0 \][/tex]
Divide every term by 2 to simplify:
[tex]\[ x^2 - 8x - 25 = 0 \][/tex]
[tex]\[ x^2 - 8x = 25 \][/tex]
#### Step 2: Complete the square
Add and subtract [tex]\(\left(\frac{-8}{2}\right)^2\)[/tex]:
[tex]\[ x^2 - 8x + 16 - 16 = 25 \][/tex]
[tex]\[ x^2 - 8x + 16 = 41 \][/tex]
[tex]\[ (x - 4)^2 = 41 \][/tex]
#### Step 3: Solve for [tex]\( x \)[/tex]
Take the square root of both sides:
[tex]\[ x - 4 = \pm \sqrt{41} \][/tex]
So,
[tex]\[ x = 4 \pm \sqrt{41} \][/tex]
Therefore, the solutions are:
[tex]\[ x_1 = 4 + \sqrt{41} \][/tex]
[tex]\[ x_2 = 4 - \sqrt{41} \][/tex]
In summary, the solutions are:
1. For the equation [tex]\( x^2 + 6x + 11 = 0 \)[/tex]:
[tex]\[ x_1 = -3 + i\sqrt{2} \][/tex]
[tex]\[ x_2 = -3 - i\sqrt{2} \][/tex]
2. For the equation [tex]\( 2x^2 - 16x = 50 \)[/tex] by completing the square:
[tex]\[ x_1 = 4 + \sqrt{41} \][/tex]
[tex]\[ x_2 = 4 - \sqrt{41} \][/tex]
### 1. Solving the quadratic equation [tex]\( x^2 + 6x + 11 = 0 \)[/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
#### Step 1: Identify coefficients
Here,
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 6 \][/tex]
[tex]\[ c = 11 \][/tex]
#### Step 2: Calculate the discriminant
The discriminant [tex]\( \Delta \)[/tex] is calculated using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values:
[tex]\[ \Delta = 6^2 - 4(1)(11) \][/tex]
[tex]\[ \Delta = 36 - 44 \][/tex]
[tex]\[ \Delta = -8 \][/tex]
Since the discriminant is negative, the equation has two complex roots.
#### Step 3: Calculate the complex roots
The roots are given by the formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since [tex]\( \Delta \)[/tex] is negative, we can write it as:
[tex]\[ \sqrt{\Delta} = \sqrt{-8} = 2i\sqrt{2} \][/tex]
Now, substituting the values:
[tex]\[ x = \frac{-6 \pm 2i\sqrt{2}}{2 \times 1} \][/tex]
[tex]\[ x = \frac{-6 \pm 2i\sqrt{2}}{2} \][/tex]
[tex]\[ x = -3 \pm i\sqrt{2} \][/tex]
So, the solutions are:
[tex]\[ x_1 = -3 + i\sqrt{2} \][/tex]
[tex]\[ x_2 = -3 - i\sqrt{2} \][/tex]
### 2. Solving [tex]\( 2x^2 - 16x = 50 \)[/tex] by completing the square
#### Step 1: Move the constant to the other side
[tex]\[ 2x^2 - 16x - 50 = 0 \][/tex]
Divide every term by 2 to simplify:
[tex]\[ x^2 - 8x - 25 = 0 \][/tex]
[tex]\[ x^2 - 8x = 25 \][/tex]
#### Step 2: Complete the square
Add and subtract [tex]\(\left(\frac{-8}{2}\right)^2\)[/tex]:
[tex]\[ x^2 - 8x + 16 - 16 = 25 \][/tex]
[tex]\[ x^2 - 8x + 16 = 41 \][/tex]
[tex]\[ (x - 4)^2 = 41 \][/tex]
#### Step 3: Solve for [tex]\( x \)[/tex]
Take the square root of both sides:
[tex]\[ x - 4 = \pm \sqrt{41} \][/tex]
So,
[tex]\[ x = 4 \pm \sqrt{41} \][/tex]
Therefore, the solutions are:
[tex]\[ x_1 = 4 + \sqrt{41} \][/tex]
[tex]\[ x_2 = 4 - \sqrt{41} \][/tex]
In summary, the solutions are:
1. For the equation [tex]\( x^2 + 6x + 11 = 0 \)[/tex]:
[tex]\[ x_1 = -3 + i\sqrt{2} \][/tex]
[tex]\[ x_2 = -3 - i\sqrt{2} \][/tex]
2. For the equation [tex]\( 2x^2 - 16x = 50 \)[/tex] by completing the square:
[tex]\[ x_1 = 4 + \sqrt{41} \][/tex]
[tex]\[ x_2 = 4 - \sqrt{41} \][/tex]
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