For the quadratic equation x² - x = 12, which of the following statements is correct?

Question

Statement #1: The equation has one solution.

Statement #2: The equation has two solutions.

Statement #3: The equation has no solutions.

(1 point)
Responses

Statement #1
Statement #1

Statement #2
Statement #2

Statement #3

Answer 1

To determine how many solutions the quadratic equation \( x^2 - x - 12 = 0 \) has, we can first rewrite the equation in standard form and then calculate the discriminant.

1. Rearranging the equation: \[ x^2 - x - 12 = 0 \]

2. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is calculated as: \[ D = b^2 - 4ac \] where \( a = 1 \), \( b = -1 \), and \( c = -12 \).

3. Substituting in the values: \[ D = (-1)^2 - 4(1)(-12) = 1 + 48 = 49 \]

The discriminant \( D = 49 \) is positive, which indicates that the quadratic equation has two distinct real solutions.

Therefore, the correct statement is:

Statement #2: The equation has two solutions.

Answer 2

Solve x2
𝑥
2
= 78
7
8
by inspection. What are the two real solutions?(1 point)
Responses

{−74, 74
−
7
4
,

7
4
}
{−74, 74
−
7
4
,

7
4
}

{−72, 72
−
7
2
,

7
2
}
{−72, 72
−
7
2
,

7
2
}

{−78, 78
−
7
8
,

7
8
}
{−78, 78
−
7
8
,

7
8
}

{−74, 7−4
−
7
4
,

7
−
4
}

Answer 3

To solve the equation \( x^2 = 78 \) by inspection, we are looking for two real numbers whose squares equal 78.

We can start by calculating the square root of 78:

\[ x = \pm \sqrt{78} \]

Now, let's break down \(\sqrt{78}\):

\[ \sqrt{78} = \sqrt{39 \cdot 2} \approx 8.83 \quad (\text{since } 8^2 = 64 \text{ and } 9^2 = 81) \]

Thus, the two real solutions are:

\[ x = \sqrt{78} \quad \text{and} \quad x = -\sqrt{78} \]

Now let’s evaluate the provided options to see which one corresponds to \( \pm\sqrt{78} \):

1. {−74, 74}
2. {−72, 72}
3. {−78, 78}
4. {−74, 7−4}

None of these clearly match \( \pm\sqrt{78} \), which is approximately \( \pm 8.83 \). It seems like the options do not include the correct solutions.

Therefore, based on solving \( x^2 = 78 \), the two real solutions are:

\[ x = -\sqrt{78} \quad \text{and} \quad x = \sqrt{78} \]

If the options are incorrect, it’s important to note that the exact values should be \( { -\sqrt{78}, \sqrt{78} } \). If you meant to refer to approximate values, it would look more like \( { -8.83, 8.83 } \).

As such, none of the given responses accurately represent the solutions to \( x^2 = 78 \).