Solved on Jan 23, 2024

Simplify the expression $9x^{2} + 30x + 25$ .

STEP 1

Assumptions
1. We are given a quadratic expression $9x^{2}+30x+25$ .
2. Our task is to simplify or factor the quadratic expression, if possible.

STEP 2

Recognize that the given expression is a quadratic trinomial of the form $ax^{2} + bx + c$ .

STEP 3

Check if the quadratic trinomial is a perfect square trinomial. A perfect square trinomial is of the form $(ax)^{2} + 2abx + b^{2}$ , which can be factored into $(ax + b)^{2}$ .

STEP 4

Identify the coefficients in the quadratic trinomial to compare with the perfect square trinomial form:
$a = 9$ , $b = 30$ , $c = 25$ .

STEP 5

Calculate $a$ and $c$ to see if they are perfect squares:
$a = 9 = (3)^{2}$ $c = 25 = (5)^{2}$

STEP 6

Since both $a$ and $c$ are perfect squares, we now check if $b$ is twice the product of the square roots of $a$ and $c$ :
$b = 2 \cdot \sqrt{a} \cdot \sqrt{c} = 2 \cdot 3 \cdot 5 = 30$

STEP 7

Since $b$ is indeed twice the product of the square roots of $a$ and $c$ , we can conclude that the quadratic trinomial is a perfect square trinomial.

STEP 8

Write the factored form of the quadratic trinomial using the square roots of $a$ and $c$ :
$9x^{2}+30x+25 = (3x + 5)^{2}$
The quadratic expression $9x^{2}+30x+25$ is factored as $(3x + 5)^{2}$ .

Was this helpful?

Simplify the expression $9x^{2} + 30x + 25$ .

STEP 1

Assumptions
1. We are given a quadratic expression $9x^{2}+30x+25$ .
2. Our task is to simplify or factor the quadratic expression, if possible.

STEP 2

Recognize that the given expression is a quadratic trinomial of the form $ax^{2} + bx + c$ .

STEP 3

Check if the quadratic trinomial is a perfect square trinomial. A perfect square trinomial is of the form $(ax)^{2} + 2abx + b^{2}$ , which can be factored into $(ax + b)^{2}$ .

STEP 4

Identify the coefficients in the quadratic trinomial to compare with the perfect square trinomial form:
$a = 9$ , $b = 30$ , $c = 25$ .

STEP 5

Calculate $a$ and $c$ to see if they are perfect squares:
$a = 9 = (3)^{2}$ $c = 25 = (5)^{2}$

STEP 6

Since both $a$ and $c$ are perfect squares, we now check if $b$ is twice the product of the square roots of $a$ and $c$ :
$b = 2 \cdot \sqrt{a} \cdot \sqrt{c} = 2 \cdot 3 \cdot 5 = 30$

STEP 7

Since $b$ is indeed twice the product of the square roots of $a$ and $c$ , we can conclude that the quadratic trinomial is a perfect square trinomial.

STEP 8

Write the factored form of the quadratic trinomial using the square roots of $a$ and $c$ :
$9x^{2}+30x+25 = (3x + 5)^{2}$
The quadratic expression $9x^{2}+30x+25$ is factored as $(3x + 5)^{2}$ .

Start learning now

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

Simplify the expression 9x2+30x+259x^{2} + 30x + 259x2+30x+25.

STEP 1

Assumptions1. We are given a quadratic expression 9x2+30x+259x^{2}+30x+259x2+30x+25.2. Our task is to simplify or factor the quadratic expression, if possible.

STEP 2

Recognize that the given expression is a quadratic trinomial of the form ax2+bx+cax^{2} + bx + cax2+bx+c.

STEP 3

Check if the quadratic trinomial is a perfect square trinomial. A perfect square trinomial is of the form (ax)2+2abx+b2(ax)^{2} + 2abx + b^{2}(ax)2+2abx+b2, which can be factored into (ax+b)2(ax + b)^{2}(ax+b)2.

STEP 4

Identify the coefficients in the quadratic trinomial to compare with the perfect square trinomial form:a=9a = 9a=9, b=30b = 30b=30, c=25c = 25c=25.

STEP 5

Calculate aaa and ccc to see if they are perfect squares:a=9=(3)2a = 9 = (3)^{2}a=9=(3)2 c=25=(5)2c = 25 = (5)^{2}c=25=(5)2

STEP 6

Since both aaa and ccc are perfect squares, we now check if bbb is twice the product of the square roots of aaa and ccc:b=2⋅a⋅c=2⋅3⋅5=30b = 2 \cdot \sqrt{a} \cdot \sqrt{c} = 2 \cdot 3 \cdot 5 = 30b=2⋅a​⋅c​=2⋅3⋅5=30

STEP 7

Since bbb is indeed twice the product of the square roots of aaa and ccc, we can conclude that the quadratic trinomial is a perfect square trinomial.

STEP 8

Write the factored form of the quadratic trinomial using the square roots of aaa and ccc:9x2+30x+25=(3x+5)29x^{2}+30x+25 = (3x + 5)^{2}9x2+30x+25=(3x+5)2The quadratic expression 9x2+30x+259x^{2}+30x+259x2+30x+25 is factored as (3x+5)2(3x + 5)^{2}(3x+5)2.

Start learning now

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

Simplify the expression $9x^{2} + 30x + 25$ .

Assumptions
1. We are given a quadratic expression $9x^{2}+30x+25$ .
2. Our task is to simplify or factor the quadratic expression, if possible.

Recognize that the given expression is a quadratic trinomial of the form $ax^{2} + bx + c$ .

Check if the quadratic trinomial is a perfect square trinomial. A perfect square trinomial is of the form $(ax)^{2} + 2abx + b^{2}$ , which can be factored into $(ax + b)^{2}$ .

Identify the coefficients in the quadratic trinomial to compare with the perfect square trinomial form:
$a = 9$ , $b = 30$ , $c = 25$ .

Calculate $a$ and $c$ to see if they are perfect squares:
$a = 9 = (3)^{2}$ $c = 25 = (5)^{2}$

Since both $a$ and $c$ are perfect squares, we now check if $b$ is twice the product of the square roots of $a$ and $c$ :
$b = 2 \cdot \sqrt{a} \cdot \sqrt{c} = 2 \cdot 3 \cdot 5 = 30$

Since $b$ is indeed twice the product of the square roots of $a$ and $c$ , we can conclude that the quadratic trinomial is a perfect square trinomial.

Write the factored form of the quadratic trinomial using the square roots of $a$ and $c$ :
$9x^{2}+30x+25 = (3x + 5)^{2}$
The quadratic expression $9x^{2}+30x+25$ is factored as $(3x + 5)^{2}$ .