Math / Algebra

Question2.6 Indicate the zeros of a function algebraicany.
The zeros of a function are the same as the $x$ -intercepts! (it's where $y=0$ ) $f(x)=0$
Directions: Find the zeros for each of the following equations.
1. $f(x)=4 x^{2}+20 x$
2. $f(x)=x^{2}+2 x-3$
3. $f(x)=5 x^{2}-5 x-150$

Studdy Solution

STEP 1

What is this asking? We need to find the $x$ values that make each function equal to zero. Watch out! Don't forget to fully factor the expressions before finding the zeros!

STEP 2

1. Find the zeros of $f(x) = 4x^2 + 20x$ .
2. Find the zeros of $f(x) = x^2 + 2x - 3$ .
3. Find the zeros of $f(x) = 5x^2 - 5x - 150$ .

STEP 3

Alright, let's set the function equal to zero: $4x^2 + 20x = 0$ .
We're doing this because the zeros of a function are the $x$ values where the function's value is zero!

STEP 4

Now, let's factor out the greatest common factor!
Looks like it's $4x$ .
So, we get $4x(x + 5) = 0$ .
Factoring helps us break down the expression into smaller, more manageable pieces.

STEP 5

Here's the exciting part!
We set each factor equal to zero and solve for $x$ .
This is because if the product of two things is zero, then at least one of them must be zero!

STEP 6

First, $4x = 0$ .
Dividing both sides by 4, we get $x = 0$ .
Boom! That's our first zero.

STEP 7

Next, $x + 5 = 0$ .
Subtracting 5 from both sides, we find $x = -5$ .
And there's our second zero!

STEP 8

Let's set our function equal to zero: $x^2 + 2x - 3 = 0$ .
Remember, zeros are where the function equals zero!

STEP 9

Time to factor!
We're looking for two numbers that multiply to $-3$ and add up to $2$ .
Those numbers are $3$ and $-1$ .
So, we have $(x + 3)(x - 1) = 0$ .

STEP 10

Now, we set each factor equal to zero and solve, just like before.

STEP 11

First, $x + 3 = 0$ .
Subtracting 3 from both sides gives us $x = -3$ .
One zero down!

STEP 12

Next, $x - 1 = 0$ .
Adding 1 to both sides, we get $x = 1$ .
We found both zeros!

STEP 13

Set the function equal to zero: $5x^2 - 5x - 150 = 0$ .
Finding those zeros!

STEP 14

Let's factor out the greatest common factor, which is 5: $5(x^2 - x - 30) = 0$ .

STEP 15

Now, we factor the quadratic inside the parentheses.
We need two numbers that multiply to $-30$ and add to $-1$ .
Those are $-6$ and $5$ .
So, we have $5(x - 6)(x + 5) = 0$ .

STEP 16

We can divide both sides by 5 to get $(x - 6)(x + 5) = 0$ .
We're dividing by 5 to simplify the equation.

STEP 17

Set each factor equal to zero and solve!

STEP 18

First, $x - 6 = 0$ .
Adding 6 to both sides gives us $x = 6$ .
Almost there!

STEP 19

Next, $x + 5 = 0$ .
Subtracting 5 from both sides, we get $x = -5$ .
We've found all the zeros!

STEP 20

The zeros for $f(x) = 4x^2 + 20x$ are $x = 0$ and $x = -5$ .
The zeros for $f(x) = x^2 + 2x - 3$ are $x = -3$ and $x = 1$ .
The zeros for $f(x) = 5x^2 - 5x - 150$ are $x = 6$ and $x = -5$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the xxx values that make each function equal to zero. Watch out! Don't forget to fully factor the expressions before finding the zeros!

STEP 2

1. Find the zeros of f(x)=4x2+20xf(x) = 4x^2 + 20xf(x)=4x2+20x.2. Find the zeros of f(x)=x2+2x−3f(x) = x^2 + 2x - 3f(x)=x2+2x−3.3. Find the zeros of f(x)=5x2−5x−150f(x) = 5x^2 - 5x - 150f(x)=5x2−5x−150.

STEP 3

Alright, let's **set** the function equal to zero: 4x2+20x=04x^2 + 20x = 04x2+20x=0.We're doing this because the *zeros* of a function are the xxx values where the function's value is **zero**!

STEP 4

Now, let's **factor** out the greatest common factor!Looks like it's 4x4x4x.So, we get 4x(x+5)=04x(x + 5) = 04x(x+5)=0.Factoring helps us break down the expression into smaller, more manageable pieces.

STEP 5

Here's the exciting part!We **set each factor equal to zero** and solve for xxx.This is because if the product of two things is zero, then at least one of them *must* be zero!

STEP 6

First, 4x=04x = 04x=0.Dividing both sides by **4**, we get x=0x = 0x=0.Boom! That's our **first zero**.

STEP 7

Next, x+5=0x + 5 = 0x+5=0.Subtracting **5** from both sides, we find x=−5x = -5x=−5.And there's our **second zero**!

STEP 8

Let's **set** our function equal to zero: x2+2x−3=0x^2 + 2x - 3 = 0x2+2x−3=0.Remember, zeros are where the function equals zero!

STEP 9

Time to **factor**!We're looking for two numbers that multiply to −3-3−3 and add up to 222.Those numbers are 333 and −1-1−1.So, we have (x+3)(x−1)=0(x + 3)(x - 1) = 0(x+3)(x−1)=0.

STEP 10

Now, we **set each factor equal to zero** and solve, just like before.

STEP 11

First, x+3=0x + 3 = 0x+3=0.Subtracting **3** from both sides gives us x=−3x = -3x=−3.One zero down!

STEP 12

Next, x−1=0x - 1 = 0x−1=0.Adding **1** to both sides, we get x=1x = 1x=1.We found both zeros!

STEP 13

**Set** the function equal to zero: 5x2−5x−150=05x^2 - 5x - 150 = 05x2−5x−150=0.Finding those zeros!

STEP 14

Let's **factor out** the greatest common factor, which is **5**: 5(x2−x−30)=05(x^2 - x - 30) = 05(x2−x−30)=0.

STEP 15

Now, we **factor** the quadratic inside the parentheses.We need two numbers that multiply to −30-30−30 and add to −1-1−1.Those are −6-6−6 and 555.So, we have 5(x−6)(x+5)=05(x - 6)(x + 5) = 05(x−6)(x+5)=0.

STEP 16

We can divide both sides by **5** to get (x−6)(x+5)=0(x - 6)(x + 5) = 0(x−6)(x+5)=0.We're dividing by 5 to simplify the equation.

STEP 17

**Set each factor equal to zero** and solve!

STEP 18

First, x−6=0x - 6 = 0x−6=0.Adding **6** to both sides gives us x=6x = 6x=6.Almost there!

STEP 19

Next, x+5=0x + 5 = 0x+5=0.Subtracting **5** from both sides, we get x=−5x = -5x=−5.We've found all the zeros!

STEP 20

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What is this asking? We need to find the $x$ values that make each function equal to zero. Watch out! Don't forget to fully factor the expressions before finding the zeros!

1. Find the zeros of $f(x) = 4x^2 + 20x$ .
2. Find the zeros of $f(x) = x^2 + 2x - 3$ .
3. Find the zeros of $f(x) = 5x^2 - 5x - 150$ .

Alright, let's set the function equal to zero: $4x^2 + 20x = 0$ .
We're doing this because the zeros of a function are the $x$ values where the function's value is zero!

Now, let's factor out the greatest common factor!
Looks like it's $4x$ .
So, we get $4x(x + 5) = 0$ .
Factoring helps us break down the expression into smaller, more manageable pieces.

Here's the exciting part!
We set each factor equal to zero and solve for $x$ .
This is because if the product of two things is zero, then at least one of them must be zero!

First, $4x = 0$ .
Dividing both sides by 4, we get $x = 0$ .
Boom! That's our first zero.

Next, $x + 5 = 0$ .
Subtracting 5 from both sides, we find $x = -5$ .
And there's our second zero!

Let's set our function equal to zero: $x^2 + 2x - 3 = 0$ .
Remember, zeros are where the function equals zero!

Time to factor!
We're looking for two numbers that multiply to $-3$ and add up to $2$ .
Those numbers are $3$ and $-1$ .
So, we have $(x + 3)(x - 1) = 0$ .

Now, we set each factor equal to zero and solve, just like before.

First, $x + 3 = 0$ .
Subtracting 3 from both sides gives us $x = -3$ .
One zero down!

Next, $x - 1 = 0$ .
Adding 1 to both sides, we get $x = 1$ .
We found both zeros!

Set the function equal to zero: $5x^2 - 5x - 150 = 0$ .
Finding those zeros!

Let's factor out the greatest common factor, which is 5: $5(x^2 - x - 30) = 0$ .

Now, we factor the quadratic inside the parentheses.
We need two numbers that multiply to $-30$ and add to $-1$ .
Those are $-6$ and $5$ .
So, we have $5(x - 6)(x + 5) = 0$ .

We can divide both sides by 5 to get $(x - 6)(x + 5) = 0$ .
We're dividing by 5 to simplify the equation.

Set each factor equal to zero and solve!

First, $x - 6 = 0$ .
Adding 6 to both sides gives us $x = 6$ .
Almost there!

Next, $x + 5 = 0$ .
Subtracting 5 from both sides, we get $x = -5$ .
We've found all the zeros!