x Substituted Evaluate True Statement?

Question

−3/4
−1/2
1/2
​For the quadratic equation −2x^2−3x+2=0 , complete the table by testing each of the given values to determine whether it is a solution. Identify which one of the values is in the solution set.​

(1 point)
Only x= ( )
is in the solution set.
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Answer 1

To determine whether each of the given values is a solution to the quadratic equation \(-2x^2 - 3x + 2 = 0\), we will substitute each value into the equation and evaluate it.

Step 1: Substitute each value into the equation

1. For \(x = -\frac{3}{4}\): \[ -2\left(-\frac{3}{4}\right)^2 - 3\left(-\frac{3}{4}\right) + 2 \] \[ = -2\left(\frac{9}{16}\right) + \frac{9}{4} + 2 \] \[ = -\frac{18}{16} + \frac{36}{16} + \frac{32}{16} \] \[ = -\frac{18}{16} + \frac{68}{16} \] \[ = \frac{50}{16} \neq 0 \]

2. For \(x = -\frac{1}{2}\): \[ -2\left(-\frac{1}{2}\right)^2 - 3\left(-\frac{1}{2}\right) + 2 \] \[ = -2\left(\frac{1}{4}\right) + \frac{3}{2} + 2 \] \[ = -\frac{1}{2} + \frac{3}{2} + 2 \] \[ = -\frac{1}{2} + \frac{3}{2} + \frac{4}{2} \] \[ = \frac{6}{2} - \frac{1}{2} \] \[ = \frac{5}{2} \neq 0 \]

3. For \(x = \frac{1}{2}\): \[ -2\left(\frac{1}{2}\right)^2 - 3\left(\frac{1}{2}\right) + 2 \] \[ = -2\left(\frac{1}{4}\right) - \frac{3}{2} + 2 \] \[ = -\frac{1}{2} - \frac{3}{2} + 2 \] \[ = -\frac{1}{2} - \frac{3}{2} + \frac{4}{2} \] \[ = \frac{4}{2} - \frac{4}{2} = 0 \]

Step 2: Fill the table and identify solutions

| x | Substituted value | Evaluate | True Statement? | |--------|-------------------------|-----------------------|----------------------| | -3/4 | -2(-3/4)² - 3(-3/4) + 2 | 50/16 \(\neq 0\) | False | | -1/2 | -2(-1/2)² - 3(-1/2) + 2 | 5/2 \(\neq 0\) | False | | 1/2 | -2(1/2)² - 3(1/2) + 2 | 0 | True |

Conclusion

Only \( x = \frac{1}{2} \) is in the solution set.

So, the answer is:

Only x= ( \(\frac{1}{2}\) ) is in the solution set.

Answer 2

Assuming an equation with one side as a squared variable expression and the other side as a numeric expression, which of the following statements is correct?

Statement #1: If the numeric expression is zero, there are two solutions.

Statement #2: If the numeric expression is zero, there is one solution.

Statement #3: If the numeric expression is zero, there are no solutions.

(1 point)
Statement # ( )
is correct

Answer 3

To analyze the statements regarding an equation with one side as a squared variable expression (let's denote it as \(x^2\)) and the other side as a numeric expression, we typically consider a form like:

\[ x^2 = k \]

Where \(k\) is a numeric expression. Now, focusing specifically on the case when the numeric expression is zero, we have:

\[ x^2 = 0 \]

Solving the equation:

To solve this equation, we take the following steps:

1. Take the square root of both sides: \[ x = \sqrt{0} \quad \text{and} \quad x = -\sqrt{0} \]

This results in: \[ x = 0 \]

Analyzing the statements:

1. Statement #1: If the numeric expression is zero, there are two solutions.

   • False: There is only one solution, \(x = 0\).

2. Statement #2: If the numeric expression is zero, there is one solution.

   • True: There is exactly one solution, \(x = 0\).

3. Statement #3: If the numeric expression is zero, there are no solutions.

   • False: There is a solution, \(x = 0\).

Conclusion:

The correct statement is:

Statement # (2) is correct.

Answer 4

Solve x^2=78 by inspection. There are two real solutions. Enter the lesser number first, and use exponents in your responses. (1 point

Answer 5

To solve the equation \( x^2 = 78 \) by inspection, we need to determine the values of \( x \) that satisfy this equation.

1. First, take the square root of both sides of the equation: \[ x = \sqrt{78} \quad \text{and} \quad x = -\sqrt{78} \]

2. The two solutions are: \[ x_1 = -\sqrt{78} \] \[ x_2 = \sqrt{78} \]

3. Since we need to enter the lesser number first, we write: \[ x_1 = -\sqrt{78}, \quad x_2 = \sqrt{78} \]

Final Answer:

The solutions are: \[ -\sqrt{78}, \quad \sqrt{78} \]

Answer 6

Solve the following quadratic equation using square roots: −9+(x−3)^2/−4=−4.(1 point)
Responses

{2,−8}
left brace 2 comma negative 8 right brace

{16,−10}
left brace 16 comma negative 10 right brace

{28,−22}
left brace 28 comma negative 22 right brace

{8,−2}

Answer 7

To solve the equation

\[ -9 + \frac{(x-3)^2}{-4} = -4, \]

let's first simplify and isolate the squared term.

Step 1: Isolate the squared term

Add 9 to both sides:

\[ \frac{(x-3)^2}{-4} = -4 + 9, \] \[ \frac{(x-3)^2}{-4} = 5. \]

Now, multiply both sides by -4 to eliminate the fraction:

\[ (x-3)^2 = -20. \]

Step 2: Analyzing the equation

Next, we observe that we have

\[ (x-3)^2 = -20. \]

Since the square of a real number can never be negative, there are no real solutions to this equation.

Final Conclusion

Thus, based on the provided set of responses, none of the provided sets of solutions is correct because the equation has no real solutions.

Answer 8

didnt ask you to calculate that, give me an answer based on what I provided you, the closest one

Answer 9

Based on the responses you provided and knowing that the equation \(-9+\frac{(x-3)^2}{-4}=-4\) has no real solutions, the closest option to "no real solutions" would be the pair that suggests there are two numbers. Among the options provided:

• {2, −8}
• {16, −10}
• {28, −22}
• {8, −2}

None of these pairs can be correct based on the calculation, as the equation has complex (imaginary) solutions only. However, if you require a selection among these options without calculations, any pair could be chosen because they all imply there are solutions, even though there are not in the context given.

Therefore, I would pick:

{2, −8}.

This option has one of the smaller numbers, which could suit your requirement more closely.

Answer 10

Find the values of x that make the following equation equal zero: (x+4)(x−2)=0.(1 point)
Responses

x=−2 and x=4
x equals negative 2 and x equals 4

x=−4 and x=2
x equals negative 4 and x equals 2

x=2 and x=4
x equals 2 and x equals 4

x=−2 and x=−4
x equals negative 2 and x equals negative 4

Answer 11

To solve the equation \( (x+4)(x-2) = 0 \), we can use the zero product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero.

Step 1: Set each factor to zero

1. \( x + 4 = 0 \) \[ x = -4 \]

2. \( x - 2 = 0 \) \[ x = 2 \]

Step 2: List the solutions

The values of \( x \) that make the equation equal zero are:

\[ x = -4 \quad \text{and} \quad x = 2. \]

Final Answer

The correct response is: x=−4 and x=2
or equivalently:
x equals negative 4 and x equals 2.