Question1. Solve each equation. a) $(2 x-5)(3 x+8)=0$

Studdy Solution

STEP 1

What is this asking? We need to find the values of $x$ that make this equation true! Watch out! Don't forget that when a product equals zero, at least one of the factors must be zero.

STEP 2

1. Apply the zero-product property
2. Solve for $x$ in each resulting equation

STEP 3

The zero-product property says that if we have two things multiplied together and they equal zero, then at least one of those things must be zero.
It's like a magic trick!
Think: if you're multiplying numbers and get zero, one of them has to be zero, right?

STEP 4

In our equation, $(2x - 5)(3x + 8) = 0$ , we have two factors: $(2x - 5)$ and $(3x + 8)$ .
So, either $2x - 5 = 0$ or $3x + 8 = 0$ .
This gives us two simpler equations to solve!

STEP 5

Our first equation is $2x - 5 = 0$ .
Let's isolate $x$ .
First, we add $5$ to both sides of the equation: $2x - 5 + 5 = 0 + 5$ , which simplifies to $2x = 5$ .
Now, we divide both sides by $2$ to get $x$ by itself: $\frac{2x}{2} = \frac{5}{2}$ .
This gives us our first solution: $x = \frac{5}{2}$ .

STEP 6

Our second equation is $3x + 8 = 0$ .
Again, we want to isolate $x$ .
We subtract $8$ from both sides: $3x + 8 - 8 = 0 - 8$ , which simplifies to $3x = -8$ .
Then, we divide both sides by $3$ : $\frac{3x}{3} = \frac{-8}{3}$ .
This gives us our second solution: $x = -\frac{8}{3}$ .

STEP 7

The solutions to the equation $(2x - 5)(3x + 8) = 0$ are $x = \frac{5}{2}$ and $x = -\frac{8}{3}$ .

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