Math / Algebra

Question5 Consider the functions $P$ and $Q$ , defined as shown. $\begin{array}{l} P(x)=x^{2}+7 x-14 \\ Q(x)=-3 x+10 \end{array}$
In the $x y$ -coordinate plane, what are the coordinates of the points at which the graphs of the equations $y=P(x)$ and $y=Q(x)$ intersect? Explain how you determined your answer. Enter your answer and your explanation in the space provided.

Studdy Solution

STEP 1

What is this asking? Where do the parabolas $y = x^2 + 7x - 14$ and $y = -3x + 10$ cross? Watch out! Don't forget there might be two crossing points!

STEP 2

1. Set up the equation
2. Solve for $x$
3. Solve for $y$

STEP 3

Alright, so we want to find where these two parabolas meet, which means they'll have the same $x$ and $y$ values at those points.
Since both equations are already solved for $y$ , let's set them equal to each other!
This gives us $x^2 + 7x - 14 = -3x + 10$ .
This is where the magic happens!

STEP 4

Now, let's rewrite the equation to make it easier to solve.
We want it to look like a standard quadratic equation, which has the form $ax^2 + bx + c = 0$ .
So, let's add $3x$ to both sides of our equation.
This gives us $x^2 + 10x - 14 = 10$ .
Remember, we're adding $3x$ to both sides to keep the equation balanced!

STEP 5

Next, let's subtract $10$ from both sides to get everything on one side.
This gives us $x^2 + 10x - 24 = 0$ .
Awesome! Now we have a nice, clean quadratic equation.

STEP 6

Time to solve for $x$ !
We can do this by factoring.
We're looking for two numbers that multiply to $-24$ and add up to $10$ .
Those numbers are $12$ and $-2$ .
So, we can rewrite our equation as $(x + 12)(x - 2) = 0$ .

STEP 7

Now, if $(x + 12)(x - 2) = 0$ , then either $x + 12 = 0$ or $x - 2 = 0$ .
This gives us two possible solutions for $x$ : $x = -12$ and $x = 2$ .
Two solutions?
That means our parabolas cross at two points!

STEP 8

We've got our $x$ values, so now we need to find the corresponding $y$ values.
Let's use the simpler equation, $y = -3x + 10$ , to do this.

STEP 9

For $x = -12$ , we have $y = -3 \cdot (-12) + 10 = 36 + 10 = 46$ .
So, one intersection point is $(-12, 46)$ .

STEP 10

For $x = 2$ , we have $y = -3 \cdot 2 + 10 = -6 + 10 = 4$ .
So, the other intersection point is $(2, 4)$ .

STEP 11

The graphs intersect at the points $(-12, 46)$ and $(2, 4)$ .

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Studdy Solution

STEP 1

What is this asking? Where do the parabolas $y = x^2 + 7x - 14$ and $y = -3x + 10$ cross? Watch out! Don't forget there might be two crossing points!

STEP 2

1. Set up the equation
2. Solve for $x$
3. Solve for $y$

STEP 3

Alright, so we want to find where these two parabolas meet, which means they'll have the same $x$ and $y$ values at those points.
Since both equations are already solved for $y$ , let's set them equal to each other!
This gives us $x^2 + 7x - 14 = -3x + 10$ .
This is where the magic happens!

STEP 4

STEP 5

Next, let's subtract $10$ from both sides to get everything on one side.
This gives us $x^2 + 10x - 24 = 0$ .
Awesome! Now we have a nice, clean quadratic equation.

STEP 6

Time to solve for $x$ !
We can do this by factoring.
We're looking for two numbers that multiply to $-24$ and add up to $10$ .
Those numbers are $12$ and $-2$ .
So, we can rewrite our equation as $(x + 12)(x - 2) = 0$ .

STEP 7

Now, if $(x + 12)(x - 2) = 0$ , then either $x + 12 = 0$ or $x - 2 = 0$ .
This gives us two possible solutions for $x$ : $x = -12$ and $x = 2$ .
Two solutions?
That means our parabolas cross at two points!

STEP 8

We've got our $x$ values, so now we need to find the corresponding $y$ values.
Let's use the simpler equation, $y = -3x + 10$ , to do this.

STEP 9

For $x = -12$ , we have $y = -3 \cdot (-12) + 10 = 36 + 10 = 46$ .
So, one intersection point is $(-12, 46)$ .

STEP 10

For $x = 2$ , we have $y = -3 \cdot 2 + 10 = -6 + 10 = 4$ .
So, the other intersection point is $(2, 4)$ .

STEP 11

The graphs intersect at the points $(-12, 46)$ and $(2, 4)$ .

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Studdy Solution

STEP 1

What is this asking? Where do the parabolas y=x2+7x−14y = x^2 + 7x - 14y=x2+7x−14 and y=−3x+10y = -3x + 10y=−3x+10 cross? Watch out! Don't forget there might be *two* crossing points!

STEP 2

1. Set up the equation2. Solve for xxx3. Solve for yyy

STEP 3

STEP 4

STEP 5

Next, let's subtract 101010 from both sides to get everything on one side.This gives us x2+10x−24=0x^2 + 10x - 24 = 0x2+10x−24=0.Awesome! Now we have a nice, clean quadratic equation.

STEP 6

Time to **solve for** xxx!We can do this by factoring.We're looking for two numbers that multiply to −24-24−24 and add up to 101010.Those numbers are 121212 and −2-2−2.So, we can rewrite our equation as (x+12)(x−2)=0(x + 12)(x - 2) = 0(x+12)(x−2)=0.

STEP 7

Now, if (x+12)(x−2)=0(x + 12)(x - 2) = 0(x+12)(x−2)=0, then either x+12=0x + 12 = 0x+12=0 or x−2=0x - 2 = 0x−2=0.This gives us two possible solutions for xxx: x=−12x = -12x=−12 and x=2x = 2x=2.Two solutions?That means our parabolas cross at *two* points!

STEP 8

We've got our xxx values, so now we need to find the corresponding yyy values.Let's use the simpler equation, y=−3x+10y = -3x + 10y=−3x+10, to do this.

STEP 9

For x=−12x = -12x=−12, we have y=−3⋅(−12)+10=36+10=46y = -3 \cdot (-12) + 10 = 36 + 10 = 46y=−3⋅(−12)+10=36+10=46.So, one intersection point is (−12,46)(-12, 46)(−12,46).

STEP 10

For x=2x = 2x=2, we have y=−3⋅2+10=−6+10=4y = -3 \cdot 2 + 10 = -6 + 10 = 4y=−3⋅2+10=−6+10=4.So, the other intersection point is (2,4)(2, 4)(2,4).

STEP 11

The graphs intersect at the points (−12,46)(-12, 46)(−12,46) and (2,4)(2, 4)(2,4).

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? Where do the parabolas $y = x^2 + 7x - 14$ and $y = -3x + 10$ cross? Watch out! Don't forget there might be two crossing points!

1. Set up the equation
2. Solve for $x$
3. Solve for $y$

Next, let's subtract $10$ from both sides to get everything on one side.
This gives us $x^2 + 10x - 24 = 0$ .
Awesome! Now we have a nice, clean quadratic equation.

Time to solve for $x$ !
We can do this by factoring.
We're looking for two numbers that multiply to $-24$ and add up to $10$ .
Those numbers are $12$ and $-2$ .
So, we can rewrite our equation as $(x + 12)(x - 2) = 0$ .

Now, if $(x + 12)(x - 2) = 0$ , then either $x + 12 = 0$ or $x - 2 = 0$ .
This gives us two possible solutions for $x$ : $x = -12$ and $x = 2$ .
Two solutions?
That means our parabolas cross at two points!

We've got our $x$ values, so now we need to find the corresponding $y$ values.
Let's use the simpler equation, $y = -3x + 10$ , to do this.

For $x = -12$ , we have $y = -3 \cdot (-12) + 10 = 36 + 10 = 46$ .
So, one intersection point is $(-12, 46)$ .

For $x = 2$ , we have $y = -3 \cdot 2 + 10 = -6 + 10 = 4$ .
So, the other intersection point is $(2, 4)$ .

The graphs intersect at the points $(-12, 46)$ and $(2, 4)$ .