Which conic is x^2 + 2x + 8y - 15 = 0 Step by Step

Step by step

The question

Question: name the conic and write in standard form x² + 2x + 8y − 15 = 0.

x² + 2x + 8y − 15 = 0

1 Name the conic

Look at the squares.

Only x² is here.

So it is a parabola.

only x² or only y² → parabola ✓

x² and y², same number in front → circle

same signs, different numbers → ellipse

different signs → hyperbola

2 Keep x on the left

Keep the x terms on the left.

Move + 8y and the plain number to the right: their signs change.

(x² + 2x) = −8y + 15

3 Complete the square for x

Half of 2 is 1.

1 times 1 is 1.

Add 1 to both sides.

(x² + 2x + 1) = −8y + 15 + 1

2 ÷ 2 = 1, 1 × 1 = 1

Careful!

What we add on the left, we add on the right too.

Then the two sides stay equal.

Careful: add 1 on both sides

4 Write as a square

The bracket is a full square: (x + 1)².

On the right, 15 plus 1 is 16.

(x + 1)² = −8y + 16

5 Take out the number in front

Take -8 out of the right side: -8(y − 2).

That is the standard form of a parabola.

(x + 1)² = −8(y − 2)

-8 × (y − 2) = −8y + 16

6 Check

Let's check with the vertex, (−1, 2).

Put it in the first equation.

(−1)² + 2 × (−1) + 8 × 2 − 15

= 1 − 2 + 16 − 15 = 0

We get 0, like the first equation says.

So the standard form is right!

Let's look back.

First: name the conic.

Then: keep x on the left.

Then: complete the square for x.

Then: write as a square.

Then: take out the number in front.

Last: check.

Answer