Question

Part A:

The quadratic equation 15x2+13x+2=0
15
𝑥
2
+
13
𝑥
+
2
=
0
can be factored into which product?

(1 point)
Responses

(3x+2)(5x+1)=0
(
3
𝑥
+
2
)
(
5
𝑥
+
1
)
=
0
open paren 3 x plus 2 close paren times open paren 5 x plus 1 close paren is equal to 0

(3x−2)(5x−1)=0
(
3
𝑥
−
2
)
(
5
𝑥
−
1
)
=
0
open paren 3 x minus 2 close paren times open paren 5 x minus 1 close paren is equal to 0

(5x+2)(3x+1)=0
(
5
𝑥
+
2
)
(
3
𝑥
+
1
)
=
0
open paren 5 x plus 2 close paren times open paren 3 x plus 1 close paren is equal to 0

(3x−2)(5x+1)=0
(
3
𝑥
−
2
)
(
5
𝑥
+
1
)
=
0
open paren 3 x minus 2 close paren times open paren 5 x plus 1 close paren is equal to 0
Question 2
Part B:

The solutions to the quadratic equation 15x2+13x+2=0
15
𝑥
2
+
13
𝑥
+
2
=
0
are

(1 point)
Responses

x=23 and x=15
𝑥
=
2
3


𝑎
𝑛
𝑑


𝑥
=
1
5
x=23 and x=15
𝑥
=
2
3


𝑎
𝑛
𝑑


𝑥
=
1
5

x=−23 and x=15
𝑥
=
−
2
3


𝑎
𝑛
𝑑


𝑥
=
1
5
x=−23 and x=15
𝑥
=
−
2
3


𝑎
𝑛
𝑑


𝑥
=
1
5

x=−23 and x=−15
𝑥
=
−
2
3


𝑎
𝑛
𝑑


𝑥
=
−
1
5
x=−23 and x=−15
𝑥
=
−
2
3


𝑎
𝑛
𝑑


𝑥
=
−
1
5

x=23 and x=−15

Answer 1

To find the factored form of the quadratic equation and its solutions, we can start with the given quadratic equation:

**Part A: Factoring the Quadratic Equation**

The quadratic equation given is:

\[ 15x^2 + 13x + 2 = 0 \]

To factor it, we look for two numbers that multiply to \(15 \cdot 2 = 30\) and add to \(13\). The numbers that satisfy this are \(3\) and \(10\).

Now we can rewrite \(13x\) as \(3x + 10x\):

\[ 15x^2 + 3x + 10x + 2 = 0 \]

Next, we group the terms:

\[ (15x^2 + 3x) + (10x + 2) = 0 \]

Factoring each group gives:

\[ 3x(5x + 1) + 2(5x + 1) = 0 \]

Now, we factor out the common factor \( (5x + 1) \):

\[ (3x + 2)(5x + 1) = 0 \]

So, the factored form of the quadratic equation is:

\[ (3x + 2)(5x + 1) = 0 \]

**Response for Part A:**

(3x + 2)(5x + 1) = 0.

---

**Part B: Finding the Solutions of the Quadratic Equation**

Now let's find the solutions using the factored form:

Setting each factor to zero:

1. \( 3x + 2 = 0 \) gives:

\[ 3x = -2 \]
\[ x = -\frac{2}{3} \]

2. \( 5x + 1 = 0 \) gives:

\[ 5x = -1 \]
\[ x = -\frac{1}{5} \]

Thus, the solutions to the quadratic equation \( 15x^2 + 13x + 2 = 0 \) are:

\[ x = -\frac{2}{3} \quad \text{and} \quad x = -\frac{1}{5} \]

**Response for Part B:**

x = -2/3 and x = -1/5.

---

In summary:
- Part A response: **(3x + 2)(5x + 1) = 0**
- Part B response: **x = -2/3 and x = -1/5**.