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

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.

3 answers

To graph the equation y=x210x24, we can rearrange it as follows:

The standard form is already given as y=x210x24, which is a downward-opening parabola.

Step 1: Find the Vertex

The vertex of a parabola given in the form y=ax2+bx+c can be found using the formula:

x=b2a

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

x=1021=102=5

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

y=(5)210(5)24=25+5024=1

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

Step 2: Finding the Roots

To find the roots of the equation x210x24=0, we can set the equation to zero.

Rearranging gives:

x210x24=0x2+10x+24=0

Now we can either factor or use the quadratic formula:

x=b±b24ac2a

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

x=10±102412421=10±100962=10±42=10±22

Calculating the two roots:

  1. x=10+22=82=4
  2. x=1022=122=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)210(7)24=49+7024=3(Point: (-7, -3))

  1. For x=3:

y=(3)210(3)24=9+3024=3(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 x210x24=0 are x=4 and x=6.

i it
a two numbers
b one point
c one equetion

It seems you are looking for a concise summary of the information related to the equation x210x24=0. Here’s a breakdown based on your specified format:

a. Two Numbers (Roots of the Equation)

The roots of the equation x210x24=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=x2+10x+24 or simply: y=x210x24

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