Which of the following is a solution to the quadratic equation −x2+x=−10? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)

Question

Which of the following is a solution to the quadratic equation −x2+x=−10? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses

x=3.7
x equals 3.7

x=0.77
x equals 0.77

x=0.27
x equals 0.27

x=1.7

GPT-4o mini · Answer

To solve the quadratic equation $-x^2 + x = -10$, we first rearrange it into standard form: $$ -x^2 + x + 10 = 0 $$ Next, we can multiply the entire equation by -1 to make the coefficient of $x^2$ positive: $$ x^2 - x - 10 = 0 $$ Now, we can use the quadratic formula to find the solutions. The quadratic formula is: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ In this case, $a = 1$, $b = -1$, and $c = -10$. Plugging these values into the formula: $$ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-10)}}{2(1)} $$ Calculating inside the square root: $$ = \frac{1 \pm \sqrt{1 + 40}}{2} = \frac{1 \pm \sqrt{41}}{2} $$ Now, we calculate $\sqrt{41}$: $$ \sqrt{41} \approx 6.403 $$ Substituting this back into our equation gives us: $$ x = \frac{1 \pm 6.403}{2} $$ This results in two solutions: 1. $$ x = \frac{1 + 6.403}{2} = \frac{7.403}{2} \approx 3.7015 \quad \text{(approximately 3.7 when rounded)} $$ 2. $$ x = \frac{1 - 6.403}{2} = \frac{-5.403}{2} \approx -2.7015 \quad \text{(approximately -2.7)} $$ From the options provided, the one that matches is: $$ x = 3.7 $$ So the solution to the quadratic equation $-x^2 + x = -10$ is: **x equals 3.7**.

adaleta · Answer

Which of the following coordinate pairs is a point on the parabola y=−4x2−53x−56?(1 point)
Responses

(−1,13)
left parenthesis neagtive 1 comma 13 right parenthesis

(−1,−7)
left parenthesis negative1 comma negative 7 right parenthesis

(−4,220)
left parenthesis negative 4 comma 220 right parenthesis

(−4,−56)

GPT-4o mini · Answer

To determine which coordinate pair is a point on the parabola defined by the equation $y = -4x^2 - 53x - 56$, we need to substitute the x-coordinates of each pair into the equation and see if we get the corresponding y-coordinates.

For the point $(-1, 13)$: \[ y = -4(-1)^2 - 53(-1) - 56 \] \[ = -4(1) + 53 - 56 = -4 + 53 - 56 = -4 + -3 = -7 \quad \text{(not 13)} \]
For the point $(-1, -7)$: \[ y = -4(-1)^2 - 53(-1) - 56 \] As calculated above: \[ = -4 + 53 - 56 = -7 \quad \text{(This is correct)} \]
For the point $(-4, 220)$: \[ y = -4(-4)^2 - 53(-4) - 56 \] \[ = -4(16) + 212 - 56 \] \[ = -64 + 212 - 56 = -64 + 156 = 92 \quad \text{(not 220)} \]
For the point $(-4, -56)$: \[ y = -4(-4)^2 - 53(-4) - 56 \] As calculated above: \[ = -64 + 212 - 56 = 92 \quad \text{(not -56)} \]

From this evaluation, the only coordinate pair that satisfies the equation $y = -4x^2 - 53x - 56$ is:

$(-1, -7)$.

adaleta · Answer

Use the image to answer the question.

A parabola is graphed on a coordinate plane. The x-axis ranges from negative 12 to 5 in increments of 1. The y-axis ranges from negative 15 to 5 in increments of 1.

Identify the vertex on the graph.

(1 point)
Responses

(−5,0)
left parenthesis negative 5 comma 0 right parenthesis

x=−5
x equals negative 5

(5,0)
left parenthesis 5 comma 0 right parenthesis

(0,−5)

GPT-4o mini · Answer

To identify the vertex of the parabola from the graph, we would typically look for the highest or lowest point on the curve, depending on whether the parabola opens upwards or downwards.

Since I don't have access to images or graphs directly, I can guide you on how to identify the vertex based on standard properties of parabolas:

For a parabola that opens upwards, the vertex is the minimum point.
For a parabola that opens downwards, the vertex is the maximum point.

If the graph shows a parabola that opens downward (which is common if the leading coefficient of the quadratic equation is negative), then the vertex will be the highest point.

Analyzing the options:

(−5, 0): This suggests a possible vertex.
x = −5: This indicates a vertical line rather than a point.
(5, 0): This is also a potential vertex.
(0, −5): This is another possible vertex.

How to determine the correct answer:

If you can visually inspect the graph, look for the point where the parabola reaches its maximum height (if it opens downwards) or minimum (if it opens upwards). The coordinates of that point will be the vertex.

If the parabola opens downward and the peak appears to be at $(-5, 0)$, that would likely be the vertex. If it opens upwards and the lowest point appears to be at $(5, 0)$, that would be the vertex.

Since I can't see the graph, I recommend checking the highest or lowest point of the parabola and comparing it to the options provided.

If you find that the vertex aligns with these coordinates based on the graph, you can select the appropriate response. If you can provide details about the parabola’s orientation (opens up or down) or any additional context from the graph, I can help narrow it down further!

adaleta · Answer

Determine the zero(s) of the quadratic function y=−5x2+16x.(1 point)
Responses

There are no zeros.
There are no zeros.

(−3.2,0)
left parenthesis negative 3.2 comma 0 right parenthesis

(0,0)(3.2,0)
left parenthesis 0 comma 0 right parenthesis left parenthesis 3.2 comma 0 right parenthesis

(0,5)(0,16)

GPT-4o mini · Answer

To determine the zero(s) of the quadratic function $y = -5x^2 + 16x$, we need to set $y$ to zero and solve for $x$. This gives us the equation:

\[ -5x^2 + 16x = 0 \]

We can factor out the common term, $x$:

\[ x(-5x + 16) = 0 \]

Now, we have a product of two factors equal to zero, which means we can set each factor equal to zero:

$x = 0$
$-5x + 16 = 0$

Now, solving for $x$ in the second equation:

\[ -5x + 16 = 0 \implies 5x = 16 \implies x = \frac{16}{5} = 3.2 \]

So, the zeros of the quadratic function $y = -5x^2 + 16x$ are:

$x = 0$
$x = 3.2$

This means the coordinate pairs corresponding to the zeros are $(0, 0)$ and $(3.2, 0)$.

Therefore, the correct response is:

(0, 0)(3.2, 0).

adaleta · Answer

Identify the vertex of y=4(x−25)2−61.(1 point)
Responses

(−100,−61)
left parenthesis negative 100 comma negative 61 right parenthesis

(25,−61)
left parenthesis 25 comma negative 61 right parenthesis

(−25,−61)
left parenthesis negative 25 comma negative 61 right parenthesis

(4,−25)

GPT-4o mini · Answer

The vertex form of a quadratic equation is expressed as:

\[ y = a(x - h)^2 + k \]

where $(h, k)$ is the vertex of the parabola.

For the given equation:

\[ y = 4(x - 25)^2 - 61, \]

we can identify $h$ and $k$:

$h = 25$
$k = -61$

Thus, the vertex of the parabola is:

\[ (25, -61). \]

Therefore, the correct response is:

(25, −61).

adaleta · Answer

What are the key points on the graph of y=x2−2x−120? Name the vertex, x-intercept(s), and y-intercept.(1 point)
Responses

x-intercepts: (−10,0) (12,0)

y-intercept: (1,−121)

vertex: (0,−120)

x -intercepts: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis y-intercept: left parenthesis 1 comma negative 121 right parenthesis vertex: left parenthesis 0 comma negative 120 right parenthesis

x-intercept: (1,−121)

y-intercept: (0,−120)

vertex: (−10,0) (12,0)

x -intercept: left parenthesis 1 comma negative 121 right parenthesis y -intercept: left parenthesis 0 comma negative 120 right parenthesis vertex: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis

x-intercepts: (−10,0) (12,0)

y-intercept: (0,−120)

vertex: (1,−121)

x -intercepts: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis y -intercept: left parenthesis 0 comma negative 120 right parenthesis vertex: left parenthesis 1 comma negative 121 right parenthesis

x-intercept: (0,−120)

y-intercept: (−10,0) (12,0)

vertex: (1,−121)

A,B,C or D

GPT-4o mini · Answer

To find the key points for the quadratic function $y = x^2 - 2x - 120$, we will calculate the vertex, x-intercepts, and y-intercept.

Step 1: Find the Vertex

The vertex of a quadratic function in standard form $ax^2 + bx + c$ can be found using the formula:

\[ h = -\frac{b}{2a} \]

For our function:

$a = 1$
$b = -2$
$c = -120$

Calculating $h$:

\[ h = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1 \]

Now, substitute $h$ back into the equation to find $k$:

\[ k = (1)^2 - 2(1) - 120 = 1 - 2 - 120 = -121 \]

So, the vertex is $(1, -121)$.

Step 2: Find the X-intercepts

To find the x-intercepts, set $y = 0$:

\[ 0 = x^2 - 2x - 120 \]

Factoring the quadratic equation:

\[ (x - 12)(x + 10) = 0 \]

Setting each factor to zero gives:

$x - 12 = 0 \implies x = 12$
$x + 10 = 0 \implies x = -10$

So the x-intercepts are $(-10, 0)$ and $(12, 0)$.

Step 3: Find the Y-intercept

To find the y-intercept, set $x = 0$:

\[ y = (0)^2 - 2(0) - 120 = -120 \]

So the y-intercept is $(0, -120)$.

Summary of Key Points

Vertex: $(1, -121)$
X-intercepts: $(-10, 0)$ and $(12, 0)$
Y-intercept: $(0, -120)$

Now, comparing this with the provided options:

Vertices: (1, -121)
X-Intercepts: (-10, 0) and (12, 0)
Y-Intercept: (0, -120)

Therefore, the correct response is:

A:

x-intercepts: $(-10, 0)$ $(12, 0)$
y-intercept: $(0, -120)$
vertex: $(1, -121)$