Graph the equation y, equals, minus, x, squared, minus, 10, x, minus, 24y=−x

Question

2
−10x−24 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the roots of the equation minus, x, squared, minus, 10, x, minus, 24, equals, 0−x
2
−10x−24=0.

Answer 1

To graph the equation $y=-x^2-10x-24$, we can rearrange it as follows:

The standard form is already given as $y=-x^2-10x-24$, which is a downward-opening parabola.

Step 1: Find the Vertex

The vertex of a parabola given in the form $y=a{x}^2+bx+c$ can be found using the formula:

$$x=-\frac{b}{2a}$$

Here, $a=-1$, $b=-10$, and $c=-24$. Plugging these values into the formula gives:

$$x=-\frac{-10}{2\cdot -1}=\frac{10}{-2}=-5$$

To find the y-coordinate of the vertex, substitute $x=-5$ into the equation:

$$y=-(-5{)}^2-10(-5)-24=-25+50-24=1$$

So, the vertex is at the point $(-5,1)$.

Step 2: Finding the Roots

To find the roots of the equation $-x^2-10x-24=0$, we can set the equation to zero.

Rearranging gives:

$$-x^2-10x-24=0⟹x^2+10x+24=0$$

Now we can either factor or use the quadratic formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Here, $a=1,b=10,c=24$:

$$x=\frac{-10\pm \sqrt{{10}^2-4\cdot 1\cdot 24}}{2\cdot 1}=\frac{-10\pm \sqrt{100-96}}{2}=\frac{-10\pm \sqrt{4}}{2}=\frac{-10\pm 2}{2}$$

Calculating the two roots:

1. $x=\frac{-10+2}{2}=\frac{-8}{2}=-4$
2. $x=\frac{-10-2}{2}=\frac{-12}{2}=-6$

Step 3: Identify Points on the Graph

Now we have the following important points:

• Vertex: $(-5,1)$
• Roots: $(-4,0)$ and $(-6,0)$

Step 4: Choose Additional Points

Let's choose two more points to fully represent the graph. We can evaluate the function at $x=-7$ and $x=-3$:

1. For $x=-7$:

$$y=-(-7{)}^2-10(-7)-24=-49+70-24=-3\text{(Point: (-7, -3))}$$

2. For $x=-3$:

$$y=-(-3{)}^2-10(-3)-24=-9+30-24=-3\text{(Point: (-3, -3))}$$

Summary of Points to Graph

The points to plot are:

• Vertex: (-5, 1)
• Root 1: (-4, 0)
• Root 2: (-6, 0)
• Point: (-7, -3)
• Point: (-3, -3)

Step 5: Graphing

You can plot these points on a Cartesian coordinate system. The parabola opens downward with roots at (-4, 0) and (-6, 0) and a vertex at (-5, 1).

Conclusion

The roots of the equation $-x^2-10x-24=0$ are $x=-4$ and $x=-6$.

Answer 2

i it
a two numbers
b one point
c one equetion

Answer 3

It seems you are looking for a concise summary of the information related to the equation $-x^2-10x-24=0$. Here’s a breakdown based on your specified format:

a. Two Numbers (Roots of the Equation)

The roots of the equation $-x^2-10x-24=0$ are:

1. $x=-4$
2. $x=-6$

b. One Point (Vertex of the Parabola)

The vertex of the parabola represented by the equation is:

• $(-5,1)$

c. One Equation

The equation to solve for the graph is: $$-y=x^2+10x+24$$ or simply: $$y=-x^2-10x-24$$

If you need any further information or assistance, feel free to ask!