Question
Use the quadratic formula to answer the questions below.
Quadratic Formula is x= −b ±b2−4ac√2a
The equation is 2x2+22x+60
Fill in the blanks below (yellow boxes).
(1 point)
Part A: GCF The GCF is = ____
The quadratic with the factored out the GCF is : ____
a = b= c=
Part B: Use the quadratic formula ___ ± ___^2 - 4(__) (__)
_____________________
2(__)
Part C: solve for x
x = (lesser number)
x = (greater number)
Quadratic Formula is x= −b ±b2−4ac√2a
The equation is 2x2+22x+60
Fill in the blanks below (yellow boxes).
(1 point)
Part A: GCF The GCF is = ____
The quadratic with the factored out the GCF is : ____
a = b= c=
Part B: Use the quadratic formula ___ ± ___^2 - 4(__) (__)
_____________________
2(__)
Part C: solve for x
x = (lesser number)
x = (greater number)
Answers
GPT-4o mini
To solve the problem using the quadratic formula, we'll work through each part step by step.
### Part A: GCF
1. **Finding the GCF**:
The given quadratic is \( 2x^2 + 22x + 60 \).
The GCF of the coefficients (2, 22, and 60) is 2.
**The GCF is = 2**
2. **Factoring out the GCF**:
We can factor out the GCF from the quadratic expression:
\[
2(x^2 + 11x + 30)
\]
**The quadratic with the factored out the GCF is: \( 2(x^2 + 11x + 30) \)**
3. **Values of a, b, and c**:
After factoring out the GCF, the coefficients for the quadratic \( x^2 + 11x + 30 \) are:
- \( a = 1 \)
- \( b = 11 \)
- \( c = 30 \)
### Part B: Use the quadratic formula
We will now calculate using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Plugging in the values from Part A:
- \( a = 1 \)
- \( b = 11 \)
- \( c = 30 \)
The formula becomes:
\[
x = \frac{-11 \pm \sqrt{11^2 - 4(1)(30)}}{2(1)}
\]
We compute:
\[
11^2 = 121
\]
\[
4(1)(30) = 120
\]
So,
\[
x = \frac{-11 \pm \sqrt{121 - 120}}{2}
\]
Thus,
\[
x = \frac{-11 \pm \sqrt{1}}{2}
\]
### Part C: Solve for x
Now, we can simplify the expression:
\[
\sqrt{1} = 1
\]
So our expression for \( x \) is:
\[
x = \frac{-11 \pm 1}{2}
\]
Calculating the two solutions:
1. For the lesser number:
\[
x = \frac{-11 - 1}{2} = \frac{-12}{2} = -6
\]
2. For the greater number:
\[
x = \frac{-11 + 1}{2} = \frac{-10}{2} = -5
\]
### Summary of Answers
**Part A:**
- GCF is = **2**
- The quadratic with the factored out the GCF is: **\( 2(x^2 + 11x + 30) \)**
- \( a = 1 \)
- \( b = 11 \)
- \( c = 30 \)
**Part B:**
- Use the quadratic formula:
\[
-11 \pm 1^2 - 4(1)(30)
\]
\[
\frac{}{} (2(1))
\]
**Part C:**
- Solve for x:
- \( x = -6 \) (lesser number)
- \( x = -5 \) (greater number)
### Part A: GCF
1. **Finding the GCF**:
The given quadratic is \( 2x^2 + 22x + 60 \).
The GCF of the coefficients (2, 22, and 60) is 2.
**The GCF is = 2**
2. **Factoring out the GCF**:
We can factor out the GCF from the quadratic expression:
\[
2(x^2 + 11x + 30)
\]
**The quadratic with the factored out the GCF is: \( 2(x^2 + 11x + 30) \)**
3. **Values of a, b, and c**:
After factoring out the GCF, the coefficients for the quadratic \( x^2 + 11x + 30 \) are:
- \( a = 1 \)
- \( b = 11 \)
- \( c = 30 \)
### Part B: Use the quadratic formula
We will now calculate using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Plugging in the values from Part A:
- \( a = 1 \)
- \( b = 11 \)
- \( c = 30 \)
The formula becomes:
\[
x = \frac{-11 \pm \sqrt{11^2 - 4(1)(30)}}{2(1)}
\]
We compute:
\[
11^2 = 121
\]
\[
4(1)(30) = 120
\]
So,
\[
x = \frac{-11 \pm \sqrt{121 - 120}}{2}
\]
Thus,
\[
x = \frac{-11 \pm \sqrt{1}}{2}
\]
### Part C: Solve for x
Now, we can simplify the expression:
\[
\sqrt{1} = 1
\]
So our expression for \( x \) is:
\[
x = \frac{-11 \pm 1}{2}
\]
Calculating the two solutions:
1. For the lesser number:
\[
x = \frac{-11 - 1}{2} = \frac{-12}{2} = -6
\]
2. For the greater number:
\[
x = \frac{-11 + 1}{2} = \frac{-10}{2} = -5
\]
### Summary of Answers
**Part A:**
- GCF is = **2**
- The quadratic with the factored out the GCF is: **\( 2(x^2 + 11x + 30) \)**
- \( a = 1 \)
- \( b = 11 \)
- \( c = 30 \)
**Part B:**
- Use the quadratic formula:
\[
-11 \pm 1^2 - 4(1)(30)
\]
\[
\frac{}{} (2(1))
\]
**Part C:**
- Solve for x:
- \( x = -6 \) (lesser number)
- \( x = -5 \) (greater number)