Question $3 y^{2}-8 y+4=5 y$

Studdy Solution

STEP 1

What is this asking? We're trying to find the mystery value(s) of $y$ that make this equation true! Watch out! Don't forget to set the equation equal to zero before factoring.
Also, remember there might be more than one solution for $y$ !

STEP 2

1. Set to Zero
2. Factor the Quadratic
3. Solve for y

STEP 3

Alright, let's kick things off by rearranging our equation to get a nice, clean zero on one side.
We can do this by subtracting $5y$ from both sides of the equation.
Remember, what we do to one side, we must do to the other to keep things balanced!
$3y^2 - 8y + 4 - 5y = 5y - 5y$ $3y^2 - 13y + 4 = 0$

STEP 4

Boom! Now we have a quadratic equation in the beautiful standard form: $ax^2 + bx + c = 0$ .
In our case, $a$ is 3, $b$ is -13, and $c$ is 4.
This setup is perfect for factoring!

STEP 5

Time to factor!
We're looking for two numbers that multiply to $a \cdot c = 3 \cdot 4 = 12$ and add up to $b = -13$ .
Let's see... $-1$ and $-12$ fit the bill perfectly!

STEP 6

Now, let's rewrite our equation using these magical numbers:
$3y^2 - 12y - y + 4 = 0$

STEP 7

Let's group the terms and factor out the greatest common factor from each group:
$3y(y - 4) - 1(y - 4) = 0$

STEP 8

Notice that $(y-4)$ is a common factor.
Let's factor that out too!
$(3y - 1)(y - 4) = 0$

STEP 9

Now, we have two factors multiplied together equaling zero.
This means either $3y - 1 = 0$ or $y - 4 = 0$ .
This is our chance to shine!

STEP 10

Let's solve $3y - 1 = 0$ . Add 1 to both sides:
$3y - 1 + 1 = 0 + 1$ $3y = 1$ Now, divide both sides by 3:
$\frac{3y}{3} = \frac{1}{3}$ $y = \frac{1}{3}$

STEP 11

Next, let's solve $y - 4 = 0$ . Add 4 to both sides:
$y - 4 + 4 = 0 + 4$ $y = 4$

STEP 12

So, the solutions to our original equation are $y = \frac{1}{3}$ and $y = 4$ !
We found two answers!
Double the fun!

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