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Math  /  Algebra

QuestionGiven the equation below, determine which type of conic section it represents, then write the equation in standard form. Your options are circle, hyperbola, ellipse, or parabola. 4547=y+644x−23x24547=y+644 x-23 x^{2}

Studdy Solution

STEP 1

What is this asking? We've got this funky equation, and we need to figure out if it's a circle, hyperbola, ellipse, or parabola, and then rewrite it in its snazziest form! Watch out! Don't get tripped up by the big numbers!
They're just there to make it look tough.
Also, remember that the order of terms in the equation doesn't tell us what kind of conic section it is.

STEP 2

1. Rewrite the equation
2. Complete the square
3. Identify the conic section
4. Write in standard form

STEP 3

Let's **group** those xx terms together to get ready for some completing-the-square action!
We'll move everything to one side to make it easier to see what we're working with. 0=−23x2+644x+y−45470 = -23x^2 + 644x + y - 4547

STEP 4

Now, let's **factor out** that −23-23 from the xx terms.
This will help us complete the square perfectly! 0=−23(x2−64423x)+y−45470 = -23(x^2 - \frac{644}{23}x) + y - 4547

STEP 5

To complete the square, we take **half** of the coefficient of our xx term, which is 12⋅(−64423)=−32223\frac{1}{2} \cdot (-\frac{644}{23}) = -\frac{322}{23}, and **square** it: (−32223)2=103684529(-\frac{322}{23})^2 = \frac{103684}{529}.
This is our **magic number**!

STEP 6

We **add** and **subtract** this magic number *inside* the parentheses to keep our equation balanced. 0=−23(x2−64423x+103684529−103684529)+y−45470 = -23(x^2 - \frac{644}{23}x + \frac{103684}{529} - \frac{103684}{529}) + y - 4547

STEP 7

Now, we can **factor** that perfect square trinomial we've created!
We'll also distribute the −23-23 to the added and subtracted magic number. 0=−23(x−32223)2+23⋅103684529+y−45470 = -23(x - \frac{322}{23})^2 + \frac{23 \cdot 103684}{529} + y - 4547 0=−23(x−32223)2+y+2384732529−45470 = -23(x - \frac{322}{23})^2 + y + \frac{2384732}{529} - 45470=−23(x−32223)2+y+2384732−24007635290 = -23(x - \frac{322}{23})^2 + y + \frac{2384732 - 2400763}{529}0=−23(x−32223)2+y−1604315290 = -23(x - \frac{322}{23})^2 + y - \frac{160431}{529}

STEP 8

Since we only have one squared term (x2x^2), we know this equation represents a **parabola**!

STEP 9

Let's **isolate** yy to get our equation into standard form. y=23(x−32223)2+160431529y = 23(x - \frac{322}{23})^2 + \frac{160431}{529}

STEP 10

The equation represents a **parabola**, and its standard form is y=23(x−32223)2+160431529.y = 23(x - \frac{322}{23})^2 + \frac{160431}{529}.

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