Find All Complex Solutions -2y^2+4y+1=-6y^2
The problem posed is asking for all complex solutions to a given quadratic equation. The equation in question can be simplified and rewritten in the standard quadratic form to make the process of finding complex roots easier. The process typically involves combining like terms and then applying the quadratic formula, which will yield solutions that might be real numbers, complex numbers, or a mix of both. The question is specifically asking for complex solutions, which indicates that even if there are real solutions, emphasis should be on identifying the solutions which are complex numbers. Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit with the property i^2 = -1.
$- 2 y^{2} + 4 y + 1 = - 6 y^{2}$
Answer
Solution:
Step 1: Rearrange the equation
Bring all terms to one side to set the equation to zero.
Step 1.1: Combine like terms
Add $6y^2$ to both sides: $-2y^2 + 4y + 1 + 6y^2 = 0$
Step 1.2: Simplify the equation
Combine $-2y^2$ and $6y^2$: $4y^2 + 4y + 1 = 0$
Step 2: Apply the quadratic formula
The solutions can be found using $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Step 3: Insert coefficients into the formula
Place $a = 4$, $b = 4$, and $c = 1$ into the quadratic formula to find $y$.
Step 4: Simplify the expression
Step 4.1: Work on the numerator
Step 4.1.1: Square the term $b$
Calculate $4^2$: $y = \frac{-4 \pm \sqrt{16 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}$
Step 4.1.2: Compute $4ac$
Multiply $-4 \cdot 4 \cdot 1$: $y = \frac{-4 \pm \sqrt{16 - 16}}{2 \cdot 4}$
Step 4.1.3: Simplify under the square root
Subtract $16$ from $16$: $y = \frac{-4 \pm \sqrt{0}}{2 \cdot 4}$
Step 4.1.4: Express $0$ as a square
Rewrite $0$ as $0^2$: $y = \frac{-4 \pm \sqrt{0^2}}{2 \cdot 4}$
Step 4.1.5: Simplify the radical
Since the square root of $0$ is $0$: $y = \frac{-4 \pm 0}{2 \cdot 4}$
Step 4.1.6: Combine terms
Adding or subtracting $0$ to/from $-4$: $y = \frac{-4}{2 \cdot 4}$
Step 4.2: Multiply the denominator
Calculate $2 \cdot 4$: $y = \frac{-4}{8}$
Step 4.3: Reduce the fraction
Step 4.3.1: Factor out common terms
Factor $4$ from $-4$: $y = \frac{4(-1)}{8}$
Step 4.3.2: Simplify the fraction
Step 4.3.2.1: Factor out $4$ from the denominator
Factor $4$ from $8$: $y = \frac{4 \cdot -1}{4 \cdot 2}$
Step 4.3.2.2: Cancel out common factors
Reduce the fraction: $y = \frac{\cancel{4} \cdot -1}{\cancel{4} \cdot 2}$
Step 4.3.2.3: Write the simplified expression
The result is: $y = \frac{-1}{2}$
Step 4.4: Position the negative sign
Place the negative sign in front: $y = -\frac{1}{2}$
Step 5: State the final solution
The solution is a repeated root: $y = -\frac{1}{2}$ (double root)
Knowledge Notes:
The problem involves solving a quadratic equation, which is an equation of the form $ax^2 + bx + c = 0$. The steps taken to solve the equation include:
Rearranging the equation to a standard form where one side equals zero.
Simplifying the equation by combining like terms.
Applying the quadratic formula, which provides the solutions to any quadratic equation and is given by $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Substituting the coefficients of the quadratic equation into the formula.
Simplifying the resulting expression, including performing operations such as squaring, multiplying, and reducing fractions.
Recognizing that the square root of zero is zero, which simplifies the expression further.
Reducing the fraction to its simplest form by canceling out common factors.
Presenting the final solution, which in this case is a repeated root, indicating that the quadratic equation has one unique solution that occurs twice.
Understanding how to manipulate and simplify algebraic expressions and apply the quadratic formula is essential in solving quadratic equations. The process also involves basic arithmetic operations and the properties of square roots and radicals.
