QuestionSolve by factoring: $2 x^{2}-2 x=-9 x-3$
Give either of the two solutions. $x=$

Studdy Solution

STEP 1

What is this asking? We need to find a value of $x$ that makes this equation true, by first rearranging the equation so it equals zero, and then factoring. Watch out! Don't forget to rearrange the equation to equal zero before factoring.
Also, remember there might be more than one solution!

STEP 2

1. Rearrange the equation
2. Factor the quadratic
3. Solve for $x$

STEP 3

First, we need to rearrange our equation to make it equal to zero.
We can do this by adding $9x$ and $3$ to both sides of the equation: $2x^2 - 2x + 9x + 3 = -9x - 3 + 9x + 3$ This simplifies to: $2x^2 + 7x + 3 = 0$ Now we have a quadratic equation set equal to zero, which is exactly what we need to start factoring!

STEP 4

We're looking for two numbers that multiply to $2 \cdot 3 = \textbf{6}$ and add up to $\textbf{7}$ .
Think about it... what two numbers multiply to 6 and add to 7?
It's $6$ and $1$ !

STEP 5

Now we rewrite the middle term ( $7x$ ) using these two numbers: $2x^2 + 6x + 1x + 3 = 0$ See how $6x + 1x$ is the same as $7x$ ?
We're just rewriting it in a clever way to help us factor.

STEP 6

Now, we can factor by grouping: $2x(x + 3) + 1(x + 3) = 0$ Notice that $(x + 3)$ is a common factor!

STEP 7

So, we factor it out: $(2x + 1)(x + 3) = 0$ Boom! We've factored our quadratic!

STEP 8

Now, we know that if two things multiplied together equal zero, then at least one of them must be zero.
So, we set each factor equal to zero: $2x + 1 = 0$ $x + 3 = 0$

STEP 9

To solve $2x + 1 = 0$ , we subtract $1$ from both sides: $2x + 1 - 1 = 0 - 1$ $2x = -1$ Then, we divide both sides by $2$ : $\frac{2x}{2} = \frac{-1}{2}$ $x = -\frac{1}{2}$

STEP 10

To solve $x + 3 = 0$ , we subtract $3$ from both sides: $x + 3 - 3 = 0 - 3$ $x = -3$

STEP 11

We found two solutions: $x = -\frac{1}{2}$ and $x = -3$ .
The problem asked for either solution, so $x = -3$ is a perfectly valid answer!

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