Math Snap

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:
$$x + 2 = 0$$ $$x = -2$$

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