Math / Algebra

QuestionSolve the polynomial inequality and graph the solution notation.
13) $x^2 - 3x - 10 \le 0$

Studdy Solution

STEP 1

What is this asking? Find the range of $x$ where $x^2 - 3x - 10$ is less than or equal to zero, and then draw a picture of that range on a number line! Watch out! Don't forget to check if the parabola opens upwards or downwards, this changes everything!
Also, remember to include the endpoints where the quadratic equals zero, since the problem says "less than or equal to zero".

STEP 2

1. Factor the quadratic
2. Find the roots
3. Determine the sign of the quadratic
4. Write the solution and graph it

STEP 3

Let's factor our quadratic expression $x^2 - 3x - 10$ .
We're looking for two numbers that multiply to $-10$ and add to $-3$ .
Those magic numbers are $-5$ and $2$ !

STEP 4

So, we can rewrite our quadratic as $(x - 5)(x + 2)$ .
Factoring helps us see where the quadratic equals zero, which is super important for solving inequalities!

STEP 5

Now, let's find where our factored quadratic $(x - 5)(x + 2)$ equals zero.
This happens when either $x - 5 = 0$ or $x + 2 = 0$ .

STEP 6

If $x - 5 = 0$ , then adding $5$ to both sides gives us $x = 5$ .
If $x + 2 = 0$ , then adding $-2$ to both sides gives us $x = -2$ .
These are our roots, also known as the zeros of the quadratic!

STEP 7

Our parabola $x^2 - 3x - 10$ opens upwards because the coefficient of $x^2$ is positive (it's $1$ ).
This means the quadratic is negative between the roots and positive outside of the roots.

STEP 8

We want to know where $x^2 - 3x - 10 \le 0$ , meaning where the quadratic is less than or equal to zero.
Since it's less than zero between the roots, our solution will be the interval between our roots, including the roots themselves!

STEP 9

Our roots are $x = -2$ and $x = 5$ .
Since we're looking for where the quadratic is less than or equal to zero, our solution is $-2 \le x \le 5$ .
This means $x$ can be any number from $-2$ to $5$ , inclusive!

STEP 10

To graph this, draw a number line, put filled-in circles at $-2$ and $5$ (because they're included in the solution), and shade the region between them.
This shaded region visually represents all the $x$ values that make our inequality true!

STEP 11

The solution to the inequality $x^2 - 3x - 10 \le 0$ is $-2 \le x \le 5$ .
The graph is a number line with closed circles at $-2$ and $5$ , and the region between them shaded.

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QuestionSolve the polynomial inequality and graph the solution notation.
13) $x^2 - 3x - 10 \le 0$

Studdy Solution

STEP 1

STEP 2

1. Factor the quadratic
2. Find the roots
3. Determine the sign of the quadratic
4. Write the solution and graph it

STEP 3

Let's factor our quadratic expression $x^2 - 3x - 10$ .
We're looking for two numbers that multiply to $-10$ and add to $-3$ .
Those magic numbers are $-5$ and $2$ !

STEP 4

So, we can rewrite our quadratic as $(x - 5)(x + 2)$ .
Factoring helps us see where the quadratic equals zero, which is super important for solving inequalities!

STEP 5

Now, let's find where our factored quadratic $(x - 5)(x + 2)$ equals zero.
This happens when either $x - 5 = 0$ or $x + 2 = 0$ .

STEP 6

If $x - 5 = 0$ , then adding $5$ to both sides gives us $x = 5$ .
If $x + 2 = 0$ , then adding $-2$ to both sides gives us $x = -2$ .
These are our roots, also known as the zeros of the quadratic!

STEP 7

Our parabola $x^2 - 3x - 10$ opens upwards because the coefficient of $x^2$ is positive (it's $1$ ).
This means the quadratic is negative between the roots and positive outside of the roots.

STEP 8

We want to know where $x^2 - 3x - 10 \le 0$ , meaning where the quadratic is less than or equal to zero.
Since it's less than zero between the roots, our solution will be the interval between our roots, including the roots themselves!

STEP 9

Our roots are $x = -2$ and $x = 5$ .
Since we're looking for where the quadratic is less than or equal to zero, our solution is $-2 \le x \le 5$ .
This means $x$ can be any number from $-2$ to $5$ , inclusive!

STEP 10

To graph this, draw a number line, put filled-in circles at $-2$ and $5$ (because they're included in the solution), and shade the region between them.
This shaded region visually represents all the $x$ values that make our inequality true!

STEP 11

The solution to the inequality $x^2 - 3x - 10 \le 0$ is $-2 \le x \le 5$ .
The graph is a number line with closed circles at $-2$ and $5$ , and the region between them shaded.

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QuestionSolve the polynomial inequality and graph the solution notation.13) x2−3x−10≤0x^2 - 3x - 10 \le 0x2−3x−10≤0

Studdy Solution

STEP 1

STEP 2

1. Factor the quadratic2. Find the roots3. Determine the sign of the quadratic4. Write the solution and graph it

STEP 3

Let's **factor** our quadratic expression x2−3x−10x^2 - 3x - 10x2−3x−10.We're looking for two numbers that **multiply** to −10-10−10 and **add** to −3-3−3.Those magic numbers are −5-5−5 and 222!

STEP 4

So, we can rewrite our quadratic as (x−5)(x+2)(x - 5)(x + 2)(x−5)(x+2).Factoring helps us see where the quadratic equals zero, which is super important for solving inequalities!

STEP 5

Now, let's find where our factored quadratic (x−5)(x+2)(x - 5)(x + 2)(x−5)(x+2) equals zero.This happens when either x−5=0x - 5 = 0x−5=0 or x+2=0x + 2 = 0x+2=0.

STEP 6

If x−5=0x - 5 = 0x−5=0, then adding 555 to both sides gives us x=5x = 5x=5.If x+2=0x + 2 = 0x+2=0, then adding −2-2−2 to both sides gives us x=−2x = -2x=−2.These are our **roots**, also known as the **zeros** of the quadratic!

STEP 7

Our parabola x2−3x−10x^2 - 3x - 10x2−3x−10 opens *upwards* because the coefficient of x2x^2x2 is positive (it's 111).This means the quadratic is *negative* *between* the roots and *positive* *outside* of the roots.

STEP 8

We want to know where x2−3x−10≤0x^2 - 3x - 10 \le 0x2−3x−10≤0, meaning where the quadratic is less than or equal to zero.Since it's less than zero *between* the roots, our solution will be the interval between our roots, including the roots themselves!

STEP 9

Our roots are x=−2x = -2x=−2 and x=5x = 5x=5.Since we're looking for where the quadratic is less than or equal to zero, our solution is −2≤x≤5-2 \le x \le 5−2≤x≤5.This means xxx can be any number from −2-2−2 to 555, inclusive!

STEP 10

To graph this, draw a number line, put filled-in circles at −2-2−2 and 555 (because they're included in the solution), and shade the region between them.This shaded region visually represents all the xxx values that make our inequality true!

STEP 11

The solution to the inequality x2−3x−10≤0x^2 - 3x - 10 \le 0x2−3x−10≤0 is −2≤x≤5-2 \le x \le 5−2≤x≤5.The graph is a number line with closed circles at −2-2−2 and 555, and the region between them shaded.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSolve the polynomial inequality and graph the solution notation.
13) $x^2 - 3x - 10 \le 0$

1. Factor the quadratic
2. Find the roots
3. Determine the sign of the quadratic
4. Write the solution and graph it

Let's factor our quadratic expression $x^2 - 3x - 10$ .
We're looking for two numbers that multiply to $-10$ and add to $-3$ .
Those magic numbers are $-5$ and $2$ !

So, we can rewrite our quadratic as $(x - 5)(x + 2)$ .
Factoring helps us see where the quadratic equals zero, which is super important for solving inequalities!

Now, let's find where our factored quadratic $(x - 5)(x + 2)$ equals zero.
This happens when either $x - 5 = 0$ or $x + 2 = 0$ .

If $x - 5 = 0$ , then adding $5$ to both sides gives us $x = 5$ .
If $x + 2 = 0$ , then adding $-2$ to both sides gives us $x = -2$ .
These are our roots, also known as the zeros of the quadratic!

Our parabola $x^2 - 3x - 10$ opens upwards because the coefficient of $x^2$ is positive (it's $1$ ).
This means the quadratic is negative between the roots and positive outside of the roots.

We want to know where $x^2 - 3x - 10 \le 0$ , meaning where the quadratic is less than or equal to zero.
Since it's less than zero between the roots, our solution will be the interval between our roots, including the roots themselves!

Our roots are $x = -2$ and $x = 5$ .
Since we're looking for where the quadratic is less than or equal to zero, our solution is $-2 \le x \le 5$ .
This means $x$ can be any number from $-2$ to $5$ , inclusive!

To graph this, draw a number line, put filled-in circles at $-2$ and $5$ (because they're included in the solution), and shade the region between them.
This shaded region visually represents all the $x$ values that make our inequality true!

The solution to the inequality $x^2 - 3x - 10 \le 0$ is $-2 \le x \le 5$ .
The graph is a number line with closed circles at $-2$ and $5$ , and the region between them shaded.