Write in Standard Form x^2+(y-3 square root of 2x)^2=1
To express the given equation, x^2 + (y - 3√(2x))^2 = 1, in Standard Form, you would need to manipulate the equation using algebraic techniques so that it resembles the general form of a conic section (such as a circle, ellipse, parabola, or hyperbola) with all its terms collected and simplified. The goal is to rearrange and simplify the terms to reveal the nature of the graph that the equation represents. The Standard Form involves having the variable terms (x and y) in specific orders and coefficients and the equation set equal to 1 or 0 depending on the type of conic section.
$x^{2} + \left(\left(\right. y - 3 \sqrt{2 x} \left.\right)\right)^{2} = 1$
Answer
Solution:
Step:1
Isolate $y$ in the equation.
Step:1.1
Remove $x^{2}$ from both sides of the given equation: $(y - 3 \sqrt{2x})^{2} = 1 - x^{2}$
Step:1.2
Extract the square root on both sides to get rid of the square on the left: $y - 3 \sqrt{2x} = \pm\sqrt{1 - x^{2}}$
Step:1.3
Simplify the square root expression $\pm\sqrt{1 - x^{2}}$.
Step:1.3.1
Express $1$ as a square: $y - 3 \sqrt{2x} = \pm\sqrt{(1)^{2} - x^{2}}$
Step:1.3.2
Apply the difference of squares identity, $a^{2} - b^{2} = (a + b)(a - b)$, where $a = 1$ and $b = x$: $y - 3 \sqrt{2x} = \pm\sqrt{(1 + x)(1 - x)}$
Step:1.4
Obtain the full solution by considering both the positive and negative roots.
Step:1.4.1
Start with the positive part of $\pm$: $y - 3 \sqrt{2x} = \sqrt{(1 + x)(1 - x)}$
Step:1.4.2
Add $3 \sqrt{2x}$ to each side: $y = \sqrt{(1 + x)(1 - x)} + 3 \sqrt{2x}$
Step:1.4.3
Now consider the negative part of $\pm$: $y - 3 \sqrt{2x} = -\sqrt{(1 + x)(1 - x)}$
Step:1.4.4
Again, add $3 \sqrt{2x}$ to each side: $y = -\sqrt{(1 + x)(1 - x)} + 3 \sqrt{2x}$
Step:1.4.5
Combine both solutions to complete the full solution: $y = \sqrt{(1 + x)(1 - x)} + 3 \sqrt{2x}$ and $y = -\sqrt{(1 + x)(1 - x)} + 3 \sqrt{2x}$
Step:2
For standard form, simplify and order terms by decreasing powers of $x$: $y = ax^{2} + bx + c$
Step:3
The standard form is thus: $y = \sqrt{(1 + x)(1 - x)} + 3 \sqrt{2x}$ and $y = -\sqrt{(1 + x)(1 - x)} + 3 \sqrt{2x}$
Knowledge Notes:
Standard Form of a Polynomial: A polynomial is in standard form when its term degrees are in descending order from left to right.
Solving Quadratic Equations: To solve equations like $(y - 3 \sqrt{2x})^{2} = 1 - x^{2}$, we often isolate the variable term, take the square root of both sides, and consider both the positive and negative solutions.
Difference of Squares: This is a common algebraic pattern where $a^{2} - b^{2} = (a + b)(a - b)$.
Square Roots: When taking the square root of both sides of an equation, we must consider both the positive and negative square roots, represented by $\pm$.
Simplifying Expressions: Involves rewriting expressions in a simpler or more convenient form, often to facilitate further operations or to clarify the structure of the expression.
