Determine whether the values -1 and 7/3 are solutions to the quadratic equation 3x^2 - 4x - 4 = 3

Question

Determine whether the values -1 and 7/3 are solutions to the quadratic equation 3x^2 - 4x - 4 = 3

Neither x = -1 nor x = 7/3 is a solution
Only x = -1 a solution
Both x = -1 and x = 7/3 are solutions
Only x = 7/3 is a solution

👇

17 answers
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Answer 1

Bot GPT 4-o mini answered
27 days ago

To determine whether the values \( x = -1 \) and \( x = \frac{7}{3} \) are solutions to the quadratic equation \( 3x^2 - 4x - 4 = 3 \), we first simplify the equation:

1. Start with the equation:
\[
3x^2 - 4x - 4 = 3
\]
2. Subtract 3 from both sides to set the equation to zero:
\[
3x^2 - 4x - 4 - 3 = 0
\]
\[
3x^2 - 4x - 7 = 0
\]

Now, we will test the given values one by one.

### Testing \( x = -1 \):
Substituting \( x = -1 \) into the left side of the equation:
\[
3(-1)^2 - 4(-1) - 7 = 3(1) + 4 - 7
\]
\[
= 3 + 4 - 7 = 0
\]

Since we get 0, \( x = -1 \) is a solution.

### Testing \( x = \frac{7}{3} \):
Substituting \( x = \frac{7}{3} \) into the left side of the equation:
\[
3\left(\frac{7}{3}\right)^2 - 4\left(\frac{7}{3}\right) - 7
\]
Calculating \( \left(\frac{7}{3}\right)^2 \):
\[
\left(\frac{7}{3}\right)^2 = \frac{49}{9}
\]
Then:
\[
3\left(\frac{49}{9}\right) - 4\left(\frac{7}{3}\right) - 7 = \frac{147}{9} - \frac{28}{3} - 7
\]
Converting \( \frac{28}{3} \) to a fraction with a denominator of 9:
\[
\frac{28}{3} = \frac{84}{9}
\]
And \( 7 = \frac{63}{9} \):
\[
\frac{147}{9} - \frac{84}{9} - \frac{63}{9} = \frac{147 - 84 - 63}{9} = \frac{0}{9} = 0
\]

Since we also get 0 here, \( x = \frac{7}{3} \) is a solution.

### Conclusion:
Both \( x = -1 \) and \( x = \frac{7}{3} \) are solutions to the equation \( 3x^2 - 4x - 7 = 0 \).

Thus, the correct answer is:
**Both x = -1 and x = 7/3 are solutions.**

Answer 2

37432 answered
27 days ago

Determine whether the values -1 and 7/3 are solutions to the quadratic equation 3x^2 - 4x - 4 = 3

Neither x = -1 nor x = 7/3 is a solution
Only x = -1 a solution
Both x = -1 and x = 7/3 are solutions
Only x = 7/3 is a solution

Answer 3

Bot GPT 4-o mini answered
27 days ago

To determine whether the values \( x = -1 \) and \( x = \frac{7}{3} \) are solutions to the quadratic equation \( 3x^2 - 4x - 4 = 3 \), we first rearrange the equation to the standard form by setting it to zero.

1. Start with the equation:
\[
3x^2 - 4x - 4 = 3
\]
2. Subtract 3 from both sides:
\[
3x^2 - 4x - 4 - 3 = 0
\]
\[
3x^2 - 4x - 7 = 0
\]

Now we will test the values \( x = -1 \) and \( x = \frac{7}{3} \) in the simplified equation \( 3x^2 - 4x - 7 = 0 \).

### Testing \( x = -1 \):
Substituting \( x = -1 \):
\[
3(-1)^2 - 4(-1) - 7
\]
Calculating:
\[
= 3(1) + 4 - 7 = 3 + 4 - 7 = 0
\]
Since the left-hand side equals 0, \( x = -1 \) is indeed a solution.

### Testing \( x = \frac{7}{3} \):
Now substituting \( x = \frac{7}{3} \):
\[
3\left(\frac{7}{3}\right)^2 - 4\left(\frac{7}{3}\right) - 7
\]
Calculating:
\[
= 3\left(\frac{49}{9}\right) - 4\left(\frac{7}{3}\right) - 7
\]
\[
= \frac{147}{9} - \frac{28}{3} - 7
\]
Convert \( \frac{28}{3} \) into ninths:
\[
\frac{28}{3} = \frac{84}{9}
\]
And also convert \( 7 \):
\[
7 = \frac{63}{9}
\]
Now, substitute back:
\[
= \frac{147}{9} - \frac{84}{9} - \frac{63}{9} = \frac{147 - 84 - 63}{9} = \frac{0}{9} = 0
\]
Since the left-hand side equals 0 here as well, \( x = \frac{7}{3} \) is also a solution.

### Conclusion:
Both \( x = -1 \) and \( x = \frac{7}{3} \) are solutions to the equation \( 3x^2 - 4x - 7 = 0 \).

Thus, the correct answer is:
**Both x = -1 and x = 7/3 are solutions.**

Answer 4

37432 answered
27 days ago

X Substituted | Evaluate | True Statement?

12
24
36
48

When set off, a certain firework follows the path of the quadratic function h = -25/36 x² +16 2/3x
h = the height of the firework in feet.
X = the horizontal distance it travels in feet.

To determine how far the firework will travel before reaching the ground, determine which value of x in table is a solution to the equation 0 = -25/36 x² +16 2/3x

24 feet
48 feet
12 feet
36 feet

Answer 5

Bot GPT 4-o mini answered
27 days ago

To determine how far the firework will travel before reaching the ground, we need to solve the equation

\[
0 = -\frac{25}{36} x^2 + 16 \frac{2}{3} x
\]

First, let's convert \( 16 \frac{2}{3} \) into an improper fraction:

\[
16 \frac{2}{3} = \frac{50}{3}
\]

So the equation can be rewritten as:

\[
0 = -\frac{25}{36} x^2 + \frac{50}{3} x
\]

To eliminate the fractions, we can multiply the entire equation by \( 36 \) (the least common multiple of the denominators):

\[
0 = -25x^2 + 600x
\]

Rearranging gives us:

\[
25x^2 - 600x = 0
\]

Factoring out \( 25x \):

\[
25x(x - 24) = 0
\]

Setting each factor equal to zero gives us:

1. \( 25x = 0 \) which leads to \( x = 0 \) (not in the provided options)
2. \( x - 24 = 0 \) which leads to \( x = 24 \)

Now, let's see which values of \( x \) from the table correspond to the height \( h \) reaching zero.

The values provided in the table are:
- 12
- 24
- 36
- 48

From our calculation, \( x = 24 \) is the only value where the firework reaches the ground (height \( h = 0 \)).

### Final Answer:
**24 feet is the solution to the equation where the height reaches zero.**

Answer 6

37432 answered
27 days ago

X Substituted | Evaluate | True Statement?

12
24
36
48

When set off, a certain firework follows the path of the quadratic function h = -25/36 x² +16 2/3x
h = the height of the firework in feet.
X = the horizontal distance it travels in feet.

To determine how far the firework will travel before reaching the ground, determine which value of x in table is a solution to the equation 0 = -25/36 x² +16 2/3x

24 feet
48 feet
12 feet
36 feet

Answer 7

Bot GPT 4-o mini answered
27 days ago

To find out how far the firework travels before reaching the ground, we need to solve the equation

\[
0 = -\frac{25}{36} x^2 + 16 \frac{2}{3} x
\]

First, let's convert \( 16 \frac{2}{3} \) into an improper fraction:

\[
16 \frac{2}{3} = \frac{50}{3}
\]

Thus, the equation becomes:

\[
0 = -\frac{25}{36} x^2 + \frac{50}{3} x
\]

Now, let's eliminate the fractions by multiplying the entire equation by \( 36 \):

\[
0 = -25x^2 + 600x
\]

Rearranging gives us:

\[
25x^2 - 600x = 0
\]

Factoring out \( 25x \):

\[
25x(x - 24) = 0
\]

Setting each factor equal to zero gives us:

1. \( 25x = 0 \) which leads to \( x = 0 \) (this is where the firework is launched, at the ground level).
2. \( x - 24 = 0 \) which leads to \( x = 24 \).

This means the firework will reach the ground again when \( x = 24 \) feet.

Now, let's check the values to confirm:

- **Substituting \( x = 12 \)**:
\[
h = -\frac{25}{36}(12)^2 + \frac{50}{3}(12)
\]
\[
= -\frac{25}{36}(144) + \frac{50}{3}(12)
= -100 + 200 = 100 \quad (\text{h is positive})
\]

- **Substituting \( x = 24 \)**:
\[
h = -\frac{25}{36}(24)^2 + \frac{50}{3}(24)
\]
\[
= -\frac{25}{36}(576) + \frac{50}{3}(24)
= -400 + 400 = 0 \quad (\text{h = 0, firework hits the ground})
\]

- **Substituting \( x = 36 \)**:
\[
h = -\frac{25}{36}(36)^2 + \frac{50}{3}(36)
\]
\[
= -\frac{25}{36}(1296) + \frac{50}{3}(36)
= -900 + 600 = -300 \quad (\text{h is negative})
\]

- **Substituting \( x = 48 \)**:
\[
h = -\frac{25}{36}(48)^2 + \frac{50}{3}(48)
\]
\[
= -\frac{25}{36}(2304) + \frac{50}{3}(48)
= -1600 + 800 = -800 \quad (\text{h is negative})
\]

### Conclusion:
The only value for \( x \) that results in \( h = 0 \) (when the firework reaches the ground) is:

**24 feet**

Answer 8

37432 answered
27 days ago

Ariel is trying to determine if x = -3 is a solution to the quadratic equation -3x^2 - 9x = 0 Which explanation demonstrates the correct reasoning? (1 point)

Answer 9

Bot GPT 4-o mini answered
27 days ago

To determine if \( x = -3 \) is a solution to the quadratic equation

\[
-3x^2 - 9x = 0,
\]

we should substitute \( x = -3 \) into the equation and evaluate.

Starting with the left side of the equation:

1. Substitute \( -3 \) into the equation:
\[
-3(-3)^2 - 9(-3).
\]

2. Calculate:
- First, \((-3)^2 = 9\)
- Then, \(-3(9) = -27\)
- Now for \(-9(-3) = 27\)

Combining those, we have:
\[
-27 + 27 = 0.
\]

Thus, substituting \( x = -3 \) results in:
\[
0 = 0,
\]
which is a true statement.

### Conclusion:
The correct explanation for whether \( x = -3 \) is a solution to the equation is:

**Yes, x = -3 is a solution because substituting it back into the equation results in the following:**
\[
X | Substituted | Evaluate | True Statement?
-3 | -3 * (-3)^2 - 9(-3) = 0| 0 = 0 | true
\]

This option correctly reflects the result of the substitution and evaluation.

Answer 10

37432 answered
27 days ago

Ariel is trying to determine if x = -3 is a solution to the quadratic equation -3x^2 - 9x = 0 Which explanation demonstrates the correct reasoning?

Answer 11

Bot GPT 4-o mini answered
27 days ago

To evaluate whether \( x = -3 \) is a solution to the quadratic equation

\[
-3x^2 - 9x = 0,
\]

we need to substitute \( x = -3 \) back into the equation and see if it satisfies it.

### Step-by-step substitution:

1. **Substituting \( x = -3 \)** into the equation:
\[
-3(-3)^2 - 9(-3).
\]

2. **Calculating each term**:
- Calculate \((-3)^2 = 9\).
- Therefore, \(-3(-3)^2 = -3(9) = -27\).
- Next, calculate \(-9(-3) = 27\).

3. **Combining the results**:
\[
-27 + 27 = 0.
\]

Thus, substituting \( x = -3 \) yields:
\[
0 = 0,
\]
which is a true statement.

### Conclusion:
Now, let's evaluate the provided options based on this substitution:

1. **Yes, x = -3 is a solution because substituting it back into the equation results in the following:**
- **X | Substituted | Evaluate | True Statement?**
- **-3 | -3 * (-3)^2 - 9(-3) = 0 | 0 = 0 | true**

This is correct.

2. **No, x = -3 is not a solution because substituting it back into the equation results in the following:**
- **X | Substituted | Evaluate | True Statement?**
- **-3 | -3 * (-3)^2 - 9(-3) = 0 | -54 ≠ 0 | false**

This is incorrect because our evaluation didn't yield \(-54\).

3. **Yes, x = -3 is a solution because substituting it back into the equation results in:**
- **X | Substituted | Evaluate | True Statement?**
- **-3 | -3 * (-3)^2 - 9(-3) = 0 | -54 = 0 | true**

This is incorrect because it claims \(-54 = 0\).

4. **No, x = -3 is not a solution because substituting it back into the equation results in:**
- **X | Substituted | Evaluate | True Statement?**
- **-3 | -3 * (-3)^2 - 9(-3) = 0 | 54 ≠ 0 | false**

This is also incorrect because we did not evaluate to \(54\).

### Final Answer:
The correct explanation demonstrating the reasoning is:

**Yes, x = -3 is a solution because substituting it back into the equation results in the following:**
\[
X | Substituted | Evaluate | True Statement?
-3 | -3 * (-3)^2 - 9(-3) = 0| 0 = 0 | true
\]

Answer 12

37432 answered
27 days ago

Show how many solutions are in the solution set for the equation 3x2 = 21.

zero
one
two
infinitely many

Answer 13

Bot GPT 4-o mini answered
27 days ago

To determine how many solutions are in the solution set for the equation

\[
3x^2 = 21,
\]

we first need to solve for \(x\).

1. **Divide both sides by 3** to simplify the equation:
\[
x^2 = \frac{21}{3} = 7.
\]

2. **Take the square root of both sides**:
\[
x = \pm \sqrt{7}.
\]

This means there are two solutions:

\[
x = \sqrt{7} \quad \text{and} \quad x = -\sqrt{7}.
\]

### Final Answer:
The equation \(3x^2 = 21\) has **two** solutions in the solution set.

Answer 14

37432 answered
27 days ago

Show how many solutions are in the solution set for the equation 3x^2 = 21.

zero
one
two
infinitely many

Answer 15

Bot GPT 4-o mini answered
27 days ago

To determine how many solutions are in the solution set for the equation

\[
3x^2 = 21,
\]

follow these steps:

1. **Divide both sides by 3** to isolate \(x^2\):
\[
x^2 = \frac{21}{3} = 7.
\]

2. **Take the square root of both sides**:
\[
x = \pm \sqrt{7}.
\]

This calculation provides two distinct solutions:
- \( x = \sqrt{7} \)
- \( x = -\sqrt{7} \)

### Conclusion:
Since there are two distinct values for \(x\), the equation \(3x^2 = 21\) has **two solutions** in the solution set.

Thus, the answer is **two**.

Answer 16

37432 answered
27 days ago

Show how many solutions are in the solution set for 2(x - 5)² + 2 = 20.

one
infinitely many
two
zero

Answer 17

Bot GPT 4-o mini answered
27 days ago

To determine how many solutions are in the solution set for the equation

\[
2(x - 5)^2 + 2 = 20,
\]

let's follow these steps:

1. **Subtract 2 from both sides** to simplify:
\[
2(x - 5)^2 = 20 - 2,
\]
\[
2(x - 5)^2 = 18.
\]

2. **Divide both sides by 2**:
\[
(x - 5)^2 = \frac{18}{2} = 9.
\]

3. **Take the square root of both sides**:
\[
x - 5 = \pm 3.
\]

This yields two equations:

1. \( x - 5 = 3 \)
2. \( x - 5 = -3 \)

4. **Solve each equation**:
- For \( x - 5 = 3 \):
\[
x = 3 + 5 = 8.
\]
- For \( x - 5 = -3 \):
\[
x = -3 + 5 = 2.
\]

### Conclusion:
The equation \(2(x - 5)^2 + 2 = 20\) has two distinct solutions: \( x = 8 \) and \( x = 2 \).

Thus, the answer is **two**.