Questionatic by factoring. $2 x^{2}+9 x-1=2 x-7$

Studdy Solution

STEP 1

What is this asking? We need to find the values of $x$ that make the equation $2x^2 + 9x - 1 = 2x - 7$ true! Watch out! Don't forget to set the equation to zero before factoring.
It's a super common mistake, so let's be extra careful!

STEP 2

1. Rewrite the equation
2. Factor the quadratic
3. Solve for x

STEP 3

First, we subtract $2x$ from both sides of the equation.
We do this to start grouping like terms together!
This gives us: $2x^2 + 9x - 2x - 1 = 2x - 2x - 7$ $2x^2 + 7x - 1 = -7$ Now, let's add 7 to both sides to completely set the equation to zero: $2x^2 + 7x - 1 + 7 = -7 + 7$ $2x^2 + 7x + 6 = 0$ Awesome! Now we have a quadratic equation set to zero, ready for factoring!

STEP 4

We're looking for two numbers that multiply to $(2 \cdot 6) = \textbf{12}$ and add up to $\textbf{7}$ .
Let's think... How about 3 and 4?
Yes, that works! $3 \cdot 4 = 12$ and $3 + 4 = 7$ .
Perfect!

STEP 5

Now, we rewrite the middle term using our magic numbers, 3 and 4: $2x^2 + 3x + 4x + 6 = 0$

STEP 6

Let's group the first two terms and the last two terms: $(2x^2 + 3x) + (4x + 6) = 0$ Now, we factor out the greatest common factor from each group.
From the first group, we can factor out $x$ , and from the second group, we can factor out 2: $x(2x + 3) + 2(2x + 3) = 0$ Notice that we have the same expression, $(2x + 3)$ , in both terms.
Let's factor that out! $(2x + 3)(x + 2) = 0$ Boom! We've factored our quadratic!

STEP 7

Now, we set each factor equal to zero and solve for $x$ .
This is because if either factor is zero, the entire expression will be zero. First factor: $2x + 3 = 0$ $2x = -3$ $x = -\frac{3}{2}$ Second factor: $x + 2 = 0$ $x = -2$

STEP 8

The solutions to the equation $2x^2 + 9x - 1 = 2x - 7$ are $x = -\frac{3}{2}$ and $x = -2$ !

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