Question $\text { x-intercept(s): }\left(\frac{q+\frac{\sqrt{10}}{z}}{z} \frac{z-\sqrt{10}}{2}, 0\right.$ --intercept: $0,3$

Studdy Solution

STEP 1

What is this asking? We need to find where this parabola crosses the x-axis and the y-axis! Watch out! Don't forget there can be two x-intercepts for a parabola.
Also, make sure to write the intercepts as coordinates!

STEP 2

1. Find the y-intercept.
2. Find the x-intercept(s).

STEP 3

The y-intercept happens when $x = 0$ .
Let's plug in $x = 0$ into our equation: $y = -2(0)^2 + 4(0) + 3$ .

STEP 4

So, $y = 3$ !
That means our y-intercept is $(0, 3)$ .
Easy peasy!

STEP 5

The x-intercept(s) happen when $y = 0$ .
So, we set our equation to zero: $0 = -2x^2 + 4x + 3$ .
Now we have a quadratic equation to solve!

STEP 6

Let's use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$ In our equation, $a = -2$ , $b = 4$ , and $c = 3$ .

STEP 7

Substitute those values into the quadratic formula: $x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot (-2) \cdot 3}}{2 \cdot (-2)}.$

STEP 8

Simplify inside the square root: $4^2$ is $16$ , and $-4 \cdot (-2) \cdot 3$ is $24$ .
So, we have $16 + 24 = 40$ inside the square root.
Our equation now looks like this: $x = \frac{-4 \pm \sqrt{40}}{-4}.$

STEP 9

We can simplify $\sqrt{40}$ as $2\sqrt{10}$ because $40 = 4 \cdot 10$ and $\sqrt{4} = 2$ .
So, we have: $x = \frac{-4 \pm 2\sqrt{10}}{-4}.$

STEP 10

We can divide both the numerator and denominator by $-2$ to simplify further: $x = \frac{2 \pm \sqrt{10}}{2}.$ This gives us two solutions for $x$ .

STEP 11

The two solutions are: $x = \frac{2 + \sqrt{10}}{2}$ and $x = \frac{2 - \sqrt{10}}{2}.$ These are the x-coordinates of our x-intercepts!

STEP 12

So, our x-intercepts are $\left(\frac{2 + \sqrt{10}}{2}, 0\right)$ and $\left(\frac{2 - \sqrt{10}}{2}, 0\right)$ .

STEP 13

The y-intercept is $(0, 3)$ .
The x-intercepts are $\left(\frac{2 + \sqrt{10}}{2}, 0\right)$ and $\left(\frac{2 - \sqrt{10}}{2}, 0\right)$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms