Solve for k 10k^2-10k+2=-K
The problem presented is a quadratic equation where you are asked to determine the value(s) of the variable \( k \). Specifically, the equation is \(10k^2 - 10k + 2 = -k\), and the goal is to solve for \( k \) by finding the roots of the equation, which may involve rearranging terms, combining like terms, and applying quadratic solution methods such as factoring, completing the square, or using the quadratic formula.
$10 k^{2} - 10 k + 2 = - K$
Answer
Solution:
Step 1:
Move $k$ to the left side by adding it to both sides of the original equation: $10k^2 - 10k + k + 2 = 0$
Step 2:
Apply the quadratic formula: $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Step 3:
Insert the coefficients $a = 10$, $b = -9$, and $c = 2$ into the quadratic formula: $k = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 10 \cdot 2}}{2 \cdot 10}$
Step 4:
Proceed with simplification:
Step 4.1:
Start with the numerator.
Step 4.1.1:
Square $-9$: $k = \frac{9 \pm \sqrt{81 - 4 \cdot 10 \cdot 2}}{20}$
Step 4.1.2:
Calculate $4 \cdot 10$: $k = \frac{9 \pm \sqrt{81 - 40 \cdot 2}}{20}$
Step 4.1.3:
Distribute the $-40$: $k = \frac{9 \pm \sqrt{81 - 80 - 40k}}{20}$
Step 4.1.4:
Subtract $80$ from $81$: $k = \frac{9 \pm \sqrt{1 - 40k}}{20}$
Step 4.1.5:
Factor out the common term of $1 - 40k$.
Step 4.1.5.1:
Factor $1$ from $1$: $k = \frac{9 \pm \sqrt{1(1) - 40k}}{20}$
Step 4.1.5.2:
Factor $-40$ from $-40k$: $k = \frac{9 \pm \sqrt{1(1) + 1(-40k)}}{20}$
Step 4.1.5.3:
Factor $1$ from $1(1) + 1(-40k)$: $k = \frac{9 \pm \sqrt{1(1 - 40k)}}{20}$
Step 4.2:
Simplify the expression under the radical: $k = \frac{9 \pm \sqrt{1(1 - 40k)}}{20}$
Step 4.3:
Reduce the fraction: $k = \frac{9 \pm \sqrt{1 - 40k}}{20}$
Step 5:
Present both solutions: $k = \frac{9 + \sqrt{1 - 40k}}{20}$ and $k = \frac{9 - \sqrt{1 - 40k}}{20}$
Knowledge Notes:
To solve a quadratic equation of the form $ax^2 + bx + c = 0$, one can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The solution involves finding the values of $a$, $b$, and $c$ from the equation, substituting them into the formula, and then simplifying the expression to find the values of $x$.
In the given problem, the equation is not initially in the standard quadratic form due to the term $-k$ on the right side. The first step is to add $k$ to both sides to obtain a quadratic equation with all terms on one side, which then allows for the application of the quadratic formula.
The simplification process involves several algebraic steps, such as squaring numbers, multiplying terms, applying the distributive property, and factoring out common terms. These steps are crucial for simplifying the expression under the square root (the discriminant) and for reducing the fraction to find the final solutions for $k$.
