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Math  /  Algebra

Questionz2+2z+2=0x=−b±b2−4ac2aa=2x2+bx+c=02c=2−b=−24ac=b2=2a=\begin{array}{c}z^{2}+2 z+2=0 \\ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \\ a=\frac{2 x^{2}+b x+c=0}{2} \quad c=2 \\ -b=-2 \quad 4 a c= \\ b^{2}=\quad 2 a=\end{array}

Studdy Solution

STEP 1

1. The given problem involves solving a quadratic equation z2+2z+2=0 z^2 + 2z + 2 = 0 .
2. The quadratic formula is provided for solving the equation.
3. The problem seems to involve identifying and using the coefficients a a , b b , and c c from the quadratic equation.

STEP 2

1. Identify the coefficients a a , b b , and c c from the quadratic equation.
2. Apply the quadratic formula to find the roots of the equation.
3. Calculate the discriminant to determine the nature of the roots.
4. Solve for the roots using the quadratic formula.

STEP 3

Identify the coefficients from the quadratic equation z2+2z+2=0 z^2 + 2z + 2 = 0 .
- The coefficient a a is the coefficient of z2 z^2 , which is 1 1 . - The coefficient b b is the coefficient of z z , which is 2 2 . - The constant term c c is 2 2 .

STEP 4

Write down the quadratic formula:
x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 5

Calculate the discriminant b2−4ac b^2 - 4ac :
- b2=22=4 b^2 = 2^2 = 4 - 4ac=4×1×2=8 4ac = 4 \times 1 \times 2 = 8 - Discriminant =b2−4ac=4−8=−4 = b^2 - 4ac = 4 - 8 = -4

STEP 6

Since the discriminant is negative, the roots are complex. Use the quadratic formula to find the roots:
z=−2±−42×1 z = \frac{-2 \pm \sqrt{-4}}{2 \times 1}

STEP 7

Simplify the expression under the square root and solve for z z :
- −4=2i \sqrt{-4} = 2i (where i i is the imaginary unit) - Substitute back into the formula:
z=−2±2i2 z = \frac{-2 \pm 2i}{2}

STEP 8

Simplify the expression to find the two roots:
- z=−2+2i2=−1+i z = \frac{-2 + 2i}{2} = -1 + i - z=−2−2i2=−1−i z = \frac{-2 - 2i}{2} = -1 - i
The roots of the equation z2+2z+2=0 z^2 + 2z + 2 = 0 are:
z=−1+iandz=−1−i z = -1 + i \quad \text{and} \quad z = -1 - i

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