To show how to solve the equation by graphing, create two functions and find their intersection points. What two functions can be used to solve the following equation, and what is their solution set?

Question

10−x2=x+4

Desmos Link

(1 point)
Responses

g(x) = x + 4, f(x) = 10+x2, {−3,2}

g(x) = x + 4, f(x) = 10+x2, {−3,2}

g(x) = x + 4, f(x) = 10+x2, {−3,1}

g(x) = x + 4, f(x) = 10+x2, {−3,1}

g(x) = x + 4, f(x) = 10−x2, {1,6}

g(x) = x

Answer 1

To solve the equation \(10 - x^2 = x + 4\) by graphing, we can rearrange it to express two functions:

1. Set one side of the equation as \(f(x)\): \[ f(x) = 10 - x^2 \]

2. Set the other side of the equation as \(g(x)\): \[ g(x) = x + 4 \]

Now, we want to find the intersection points of the two functions \(f(x)\) and \(g(x)\). This can be done by finding where \(f(x) = g(x)\).

To find the solution set, we can solve for the points where these two functions intersect:

Setting the functions equal to each other:

\[ 10 - x^2 = x + 4 \]

Rearranging the equation:

\[ 10 - 4 = x + x^2 \] \[ 6 = x + x^2 \] \[ x^2 + x - 6 = 0 \]

Factoring the quadratic:

\[ (x - 2)(x + 3) = 0 \]

Finding the roots:

1. \(x - 2 = 0 \Rightarrow x = 2\)
2. \(x + 3 = 0 \Rightarrow x = -3\)

Conclusion:

The intersection points (solutions) of the original equation \(10 - x^2 = x + 4\) are:

• \(x = 2\)
• \(x = -3\)

Thus, the solution set is \({-3, 2}\).

Answer:

The correct response is: g(x) = x + 4, f(x) = 10 - x^2, {−3, 2}

Answer 2

What function should be graphed in order to solve the equation 9x−6 =−8

Desmos Link

(1 point)
Responses

f(x) = 9x + 2
f(x) = 9x + 2

f(x) = 9x - 14
f(x) = 9x - 14

f(x) =17x - 6
f(x) =17x - 6

f(x) =x -6

Answer 3

To solve the equation \(9x - 6 = -8\) by graphing, we can rearrange the equation into the form of a function set equal to \(y\).

1. Start with the original equation: \[ 9x - 6 = -8 \]

2. Add 6 to both sides to isolate the \(x\) term: \[ 9x = -8 + 6 \] \[ 9x = -2 \]

3. Divide both sides by 9: \[ x = -\frac{2}{9} \]

Now, to represent this equation as a function, we can rewrite it in function form:

Set the left-hand side equal to \(f(x)\): \[ f(x) = 9x - 6 \]

To find the solution, we can also set a constant function equal to the right-hand side:

\[ g(x) = -8 \]

Conclusion:

You should graph the function \(f(x) = 9x - 6\) and the horizontal line \(g(x) = -8\) to find their intersection point, which will indicate the solution to the equation.

Answer:

The correct response is: f(x) = 9x - 6

Answer 4

wrong

Answer 5

I apologize for the oversight. Let's carefully solve the equation \(9x - 6 = -8\) and identify the correct function to graph:

Starting from the equation:

\[ 9x - 6 = -8 \]

Rearranging the Equation:

1. Add 6 to both sides: \[ 9x = -8 + 6 \] \[ 9x = -2 \]

2. Divide both sides by 9: \[ x = -\frac{2}{9} \]

Functions to Graph:

To graph this equation, we can set one side equal to a function. The original equation can be manipulated in various forms:

1. You can keep the left side as: \[ f(x) = 9x - 6 \] And set this equal to a constant: \[ g(x) = -8 \]

However, that was not what I was looking for based on your feedback. Let's express it to find a function only for the left-hand side equal to 0:

• Rearranging the original equation gives us: \[ 9x - 6 + 8 = 0 \quad \Rightarrow \quad 9x + 2 = 0 \] Thus, we can define a function based on this: \[ f(x) = 9x + 2 \]

Conclusion:

The function you should graph in order to solve the equation \(9x - 6 = -8\) would then be:

f(x) = 9x + 2

Again, I'm sorry for the confusion, and thank you for your patience!

Answer 6

What is the solution to the equation 2x - 1 = 4x + 3?

Answer 7

To solve the equation \(2x - 1 = 4x + 3\), follow these steps:

1. Get all terms involving \(x\) on one side: Subtract \(2x\) from both sides: \[ -1 = 4x - 2x + 3 \] This simplifies to: \[ -1 = 2x + 3 \]

2. Isolate the \(x\) term: Subtract 3 from both sides: \[ -1 - 3 = 2x \] This simplifies to: \[ -4 = 2x \]

3. Solve for \(x\): Divide both sides by 2: \[ x = -2 \]

Conclusion

The solution to the equation \(2x - 1 = 4x + 3\) is: \[ \boxed{-2} \]