Question

Part 1: Solve each of the quadratic equations below. Show your work.

x2 - 16 = 0 and x2 = -2x + 24

Part 2: Describe what the solution(s) represent to the graph of each.

Part 3: How are the graphs alike? How are they different?

Answer 1

I assume the x2 means \[x^2\]?

Answer 2

(x+4)(x-4)=0, roots of graph are +4 and -4

Answer 3

(x-4)(x+6)=0, roots of graph are +4 and -6

since the co-efficient of each graph is > 0 (ax^2+bx+c=0 where a>0) both graphs will be of a minimum parabola, u shape, and will share an common root of -4

Answer 4

First Equation:
\[x^{2}-16=0\]
Add 16 to both sides:
\[x^2=16\]
Take the square root of both sides:
\[x=\sqrt{16}=\pm4\]
It's +/- 4 because if you square -4 you still get 16 as you would with +4.

Second Equation:
\[x^2 = -2x + 24\]
With this one I would do what's called "completing the square":

First get all x terms on one side, leaving a space between the x term and the equals sign:
\[x^2+2x+\left[ ? \right]=24\]
Then, you figure out what "completes the square" by taking the coefficient of the x term (in this case 2), halving it, and then squaring it:
\[(1/2)*2=1\rightarrow1^2=1\]
So the question mark in the previous equation is replaced by 1. But, you can't just stick a 1 into the equation without canceling it out somehow. We do this by also adding 1 to the right side of the equation, and then simplifying and factoring the new equation:
\[x^2+2x+1=24+1\rightarrow (x+1)^{2}=25\]
Finally, we solve for x:

Take the square root of both sides:
\[\sqrt{(x+1)^2}=\sqrt{25}\rightarrow x+1=\pm5\]
Subtract 1 from both sides, using both +5 and -5:
\[x+1-1=5-1=4\]
\[x+1-1=-5-1=-6\]
So x = +4 and -6 in this equation.

PART 2:
So the first equation with the solution, x=+4,-4, its graph will have x-intercepts of 4 and -4, and since the x^2 term is positive, it will open in the positive y-direction (U shape).

For the second equation with the solution, x=+4,-6, its graph will have x-intercepts of 4 and -4, and since the x^2 term is positive, it will open in the positive y-direction (U shape).

PART 3:
The graph of equation 2 will be a bit wider than that of equation 1 since its x-intercepts are farther apart. However, both equations' graphs share an x-intercept of 4.

Hope that more fully answered it