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

Question

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 1

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

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

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

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

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

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

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

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)

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

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

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| -54 = 0 | true

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

Answer 9

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

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?

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

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

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| -54 = 0 | true

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

Answer 11

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

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

Answer 13

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

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

zero
one
two
infinitely many

Answer 15

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

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

Answer 17

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. \( x - 5 = -3 \)

3. 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.