Alg2 2.7 QUADRATIC System of EQUATIONS

2.7 Quadratic System of Equations

Labelling

Label the key pieces of a quadratic graph.

Drag labels to their correct position on the image

Quadrat

Roots

Y-intercept

X-Intercept

Zero

Vertex

Let's recap this in Vertex Form

Let's begin by labeling important pieces of the equation!!!

What is the vertex coordiante point?

Is it a Maximum or Minimum?

Drag and Drop

Drag and drop the correct labels of the following equation: $$f\left(x\right)=3\left(x+6\right)^2-4$$ ​ ​

Placeholder

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Placeholder

Drag these tiles and drop them in the correct blank above

Minimum

(-6,-4)

(6,-4)

Maximum

Drag and Drop

Drag and drop the correct labels of the following equation: $$f\left(x\right)=-4\left(x-10\right)^2$$ ​ ​

Placeholder

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Placeholder

Drag these tiles and drop them in the correct blank above

Maximum

(10,0)

Minimum

(6,-4)

Multiple Select

What can the roots to a quadratic equation be called?

• zeros

• solutions

• x-intercepts

• none of the choices

Multiple Choice

What are the x- intercepts?

• x= 0 and x= -4

• x= 0 and x= 4

• y= 0

• x= 2

Multiple Choice

Solve the following quadratic equation: $$5x^2=10x$$  

• $$x=-2$$  

• $$x=2$$  

• $$x=2$$  & $$x=0$$  

• $$x=-2$$  & $$x=0$$  

Multiple Choice

Solve for the values of x:
(x + 2)(x - 3) = 0

• { 2, 3 }

• { -2, -3 }

• { 2, 3 }

• { -2, 3 }

Multiple Choice

Solve Using the Quadratic Formula
 x² + 4x - 40 = -8

• -10 & -4

• -4 & 10

• -8 & 4

• 8 & -4

Multiple Choice

What would this quadratic function look like?

$$y=-2x^{2\ }+3x\ +4$$  

​Let's take a look...

​What did we do previously if we wanted to find out the value of x, when y = 0, i.e.

​y = 0 , x = ?

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​Here's a question....

Multiple Choice

To find out the value of x, when y = 0, we had to:

• Draw the line, $$x\ =\ 0$$  

• Draw the line,  $$x\ =\ 2$$  

• Draw the line, $$y\ =\ 0$$  

• Draw the line,

  $$y=2$$  

​Yes, we drew the line

​This is a special case, because y = 0 is actually coincides with the x-axis.

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Therefore, these values of x are called the x - intercepts.

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Fill in the Blanks

​Now, let us review what we did...

​For the quadratic function,

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​When we wanted to find the x values, ( x = ? ) we drew the function,

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If we combined the two functions above, we get a quadratic equation.

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​Let's recap a bit more...

Whenever we have an equation, the two sides must equal. So for the quadratic equation,

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​what do you think would be the values of x that would make the equation true?

Have you guessed it?

​Yes, the values that we got just now...

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these are called the solutions or the roots ​make the equation below true.

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Try it out...

Multiple Choice

So now, if i wanted to find the roots for the quadratic equation,  $$-2x^2+3x+4\ =\ 3$$   what would you do?

• Reuse the function  $$y=-2x^2+3x+4$$  and draw the function  $$y=3$$  and find the intercepts.

• Panic!

Fill in the Blanks

Multiple Choice

Convert the following quadratic equation to standard form.

$$4x^2+21x=-6$$  

• $$4x^2+21x-6=0$$  

• $$4x^2+21x+6=0$$  

Multiple Choice

What is the "c" term of the following quadratic equation:

$$2x^2+5x=0$$  

• -6

• 5

• 0

• 6

Multiple Choice

What is the first step when solving a quadratic equation by factoring?

• Put the equation in standard form

• Factor

• Set the factors equal to zero and solve.

Writing Quadratic Equations

1. Identifying the vertex

2. Solving for a

3. Writing in vertex form

4. Changing to standard form

Multiple Choice

Whats the vertex of $$f\left(x\right)=\left(x-2\right)^2-5$$  

• (2,-5)

• (2,5)

• (-2,-5)

• (-2,5)

Multiple Choice

What is the vertex of $$f\left(x\right)=\left(x+4\right)^2$$  

• (0,4)

• (4,0)

• (-4,0)

• (0,-4)

Multiple Choice

Which of the following is the correct form for vertex form of a quadratic?

• $$x=a\left(y-h\right)^2+k$$  

• $$y=a\left(x-h\right)^2+k$$  

• $$y=a\left(x-k\right)^2+h$$  

• $$y=mx+b$$  

Multiple Choice

Identify the a, h, and k term from the following equation: $$\left(x+3\right)^2+2$$  

• a=-1, h=3, k=2

• a=3, h=3, k=3

• a=1, h=-3, k=2

• a=1, h=-3, k=-2

Multiple Choice

Determine if the following equation has a maximum or minimum: $$-\left(x-2\right)^2+7$$  

• Maximum

• Minimum

Multiple Choice

Given the equation: y = -(x - 4)² - 3,

does this parabola have a maximum or minimum value?

• Maximum

• Minimum

Match

Match the following:

Vertex $$\left(1,\ 6\right)$$ going through the point $$\left(-2,\ 0\right)$$

• h

• k

• x

• y

1

6

-2

0

• 1

• 6

• -2

• 0

Multiple Choice

Vertex $$\left(1,6\right)$$ going through point $$\left(-2,0\right)$$

Which of the following shows the values correctly substituted into the vertex form?

• $$0=a\left(-2-1\right)^2+6$$

• $$6=a\left(-2-1\right)^2+0$$

• $$0=a\left(1+2\right)^2+6$$

• $$6=a\left(1+2\right)^2+0$$

Multiple Choice

Vertex $$\left(1,6\right)$$ going through point $$\left(-2,0\right)$$ .

What is the value of a?

• $$3$$

• $$-\frac{2}{3}$$

• $$-\frac{3}{2}$$

• $$15$$

Math Response

Vertex $$\left(1,6\right)$$ going through point $$\left(-2,0\right)$$ .

What is the equation in standard form?

Use y= instead of f(x)=

Type answer here

Correct Answer: y=-\frac{2}{3}\left(x-1\right)^2+6

Deg^°

Rad

Multiple Choice

Is (2, 3) part of the solution set? 
y < 2x² -3x+1

• Yes
• No

Linear-Quadratic Systems can have the following types of solutions.

Two Solutions: (0,-2) and (3,1)

• System of equations cross at two points on the graph

One Solution: (1,-3)

• System of equations meet at one point on the graph

No Solution

This means that both equations NEVER intersect

Multiple Choice

If the a quadratic equation crosses the linear equation at two points on a graph, then the system of equations has?

• No solution

• One solution

• Two solution

• All real numbers

Multiple Choice

A linear-quadratic system touch at the point (1,-3) is said to have how many solutions?

• Two solutions

• One solution

• No solution

• Many solutions

Multiple Choice

Which of the following represents the first step in solving:

y = x² + 3x - 5

y = x + 3

• x² - 3x + 5 = x - 3

• x² + 3x - 5 = x + 3

• x² + 3x - 5 + x + 3 = 0

• x² + 3x - 5 = 0

Multiple Choice

Find the solution.

• (2,3) and (3,4)
• (-3,-15) and (4,-8)
• (-3,10) and (8,4)
• (-8,3) and (-4,10)

Multiple Choice

Which inequality is represented by the graph?

• y > (x - 3)² - 3

• y ≥ (x + 3)² - 3

• y > (x + 3)² - 3

• y ≥ (x - 3)² - 3

Multiple Choice

Find the solution:
y = x² - 4x + 6
y = x + 2

• (3, 1) and (6, 4)
• (-1, 1) and (4, 2)
• (1, 3) and (4, 6)
• (1, 4) and (3, 6)

Multiple Choice

What are the solutions to the system:
y = x² + 3x - 5
y = x + 3

• (-4, 2) and (-1, 5)
• (-4, -1) and (2, 5)
• (2, -1 and (-4, 5)
• (-4, 1) and (2, -1)

Multiple Choice

Find the solution.

• (0,7) and (0,-3)
• (2,6) and (7,4)
• (3,0) and (7,4)
• (-1,5) and (-3,9)

Multiple Choice

• y ≥ -(x + 4)² + 1
• y < -(x - 4)² + 1
• y ≤ (x - 4)² + 1
• y ≤ -(x - 4)² + 1

Multiple Choice

How many solutions does this system of equations have?

• one solution

• two solutions

• no solutions

• All of the above

Multiple Choice

Find the solution:
y = x² - 4x + 6
y = x + 2

• (3, 1) and (6, 4)
• (-1, 1) and (4, 2)
• (1, 3) and (4, 6)
• (1, 4) and (3, 6)

Multiple Choice

How many solutions does this system of equations have?

• one solution

• two solutions

• three solutions

• no solutions

Multiple Choice

Determine the solution of the given quadratic inequality.

$$x\left(4x-5\right)\le0$$  

• $$x\le0\ or\ x\ge\frac{5}{4}$$  

• $$0\le x\le\frac{5}{4}$$  

Multiple Choice

Solve the system of Equations:

y= 3

y= x² - 4x + 7

• (2,3) and (4,3)

• (3,2) and (3,4)

• Only (-2,3)

• Only (2,3)

Multiple Choice

Determine the solution to the given linear-quadratic function.

• (0, 0)
• (3, 0)
• (2, -3)
• (-3, 2)

Multiple Choice

Graph the following equations:

y= - 2x - 3

y=x²-10x +22

Multiple Choice

Determine the solution of the given quadratic inequality.

$$\left(3x-9\right)\left(x-10\right)\ge0$$  

• $$x\le3\ or\ x\ge10$$  

• $$3\le x\le10$$  

Multiple Choice

Solve the depicted system algebraically. Change to y = form.

• (0,-7) and (1, 6)

• (0,-7) and (1, -6)

• (0, 7) and (1, -6)

• (-7,0) and (1, -6)

Multiple Choice

Solve the system of linear and quadratic equations using substitution:
y = 5
y = 2x² - 16x + 29

• (6, 5) and (2, 5)
• (2, 6)
• (6, 2)
• (5, 6) and (5, 2)

Multiple Choice

Determine the solution to the given linear-quadratic function.

• (0, 0)
• (3, 0)
• (2, -3)
• (-3, 2)

Multiple Select

The following symbols are used to denote inequality except _______________________.

• >

• =

• <

• $$\ge$$

Multiple Choice

Given the system of equations:
y = x² + 3x - 5
y = x + 3
When x = -4, what is the value of y?

• -1
• 1
• 7
• -33

Multiple Choice

Determine the number of solutions to the given linear-quadratic system.

• 0 solutions
• 1 solution
• 2 solutions
• 3 solutions

Multiple Choice

The system of a linear and a quadratic function can have the following number of solutions, EXCEPT

• No solution
• 1 solution
• 2 solutions
• more than 2 solutions

Multiple Select

Which of the following is an example of quadratic inequality?

• 2x² + 3x + 4 > 0

• 15x ≤ 10

• x² - x +15 < -30

• x² + 4x - 5 = 0

Multiple Choice

Solve the System Algebraically.

$$y=-\left(x-2\right)^2+4$$

  $$y=-5$$  

• (-1,-5)

• (5,-5)

• (-1,-5) and (5,-5)

• (-5,5) 

Multiple Choice

Which of the following graphs correctly represents:

y=x²

y=-2x+1

Multiple Choice

Given the system of equations:
y = x² + 3x - 5
y = x + 3
When x = -4, what is the value of y?

• -1
• 1
• 7
• -33

Multiple Choice

Which of the following is the first step in solving quadratic inequality?

• Shade the interval on the number line that represents the solution of the quadratic inequality

• Plot the roots in a number line.

• Change the inequality symbol to equality symbol.

• Solve for the roots of the quadratic equation by factoring.

Multiple Choice

Determine the number of solutions to this system of equations.

• No solution

• 1 solution

• 2 solutions

• infinite solutions

Multiple Choice

Solve the System Algebraically.

$$y=-\left(x-2\right)^2+4$$

  $$y=-5$$  

• (-1,-5)

• (5,-5)

• (-1,-5) and (5,-5)

• (-5,5) 

Multiple Choice

Determine the number of solutions to this system of equations.

• No solution

• 1 solution

• 2 solutions

• infinite solutions

Multiple Choice

What are the solutions to the graphed system?

• (0,-2) and (-3,1)

• (-2,0) and (2,0)

• (0,-2) and (3,1)

• (2,2) and (1,0)

Multiple Choice

Which of the following odes not belong to the group?

• x² - 3x + 4 > 0

• 2x² > 4

• x² + 6x < 3

• x³ + 4x + 8 < 0

Multiple Choice

What is the solution?

• (1, -1)
• (-1, 1)
• (0, -2)
• (0, 1)

Multiple Choice

Give the correct factor of x² -12x + 27.

• (x + 3) (x + 9)

• (x - 3) (x - 9)

• (x - 3) (x + 9)

• (x + 3) (x - 9)

Multiple Choice

Which of the following is the correct interval of x² + 2x -24 > 0

• x > -6

• x > 4

• x < 4

• 4 < x < -6

Multiple Choice

Solve the quadratic inequality:

$$x^2-4x-5\le0$$  

• $$-1\le x\le5$$  

• $$-5\le x\le1$$  

• $$x\ge5\ ,\ \ x\le-1$$  

• $$x\le-5\ ,\ \ \ x\ge1$$  

Multiple Choice

Solve the quadratic inequality:

$$x^2+9x+18<0$$  

• $$x<-6\ ,\ \ x>-3$$  

• $$-6<x<-3$$  

• $$x<3\ ,\ \ x>6$$  

• $$3<x<6$$  

Multiple Choice

Solve the quadratic inequality:

$$x^2+11x+10\ge0$$  

• $$x\le-10\ ,\ \ x\ge-1$$  

• $$x\ge-10\ ,\ \ x\le-1$$  

• $$-10\le x\le-1$$  

• $$x\le1,\ x\ge10$$  

Multiple Choice

Solutions are in the ___________ area.

• non-shaded

• shaded

• vertex

• axis of symmetry

Multiple Choice

Which correctly shows the inequality $$y<\left(x+4\right)^2+1$$  ?

Multiple Choice

Which correctly shows the inequality $$y>x^2+4x+6$$  ?

Multiple Choice

Is the point (2, 1) a solution to the inequality shown?

• Yes

• No

Multiple Choice

Is the point (2, 2) a solution to the inequality $$y\ge-2\left(x-3\right)^2+2$$  

• Yes

• No

Multiple Choice

Which correctly shows the inequality $$y\ge-\left(x+2\right)^2-4$$  ?

Multiple Choice

Which of the following graphs below represents the solution to the system of quadratic equations?

Explanation Slide...

Pay attention to the difference between < and <= :) Remember that < and > indicate a dashed line whereas <= and >= results in solid lines.

Multiple Select

Identify the solutions to the system of quadratic inequalities. Select all that apply.

• {(0, 1)}

• {(1, 0)}

• {(-1, 0)}

• {(0, -1)}

Multiple Choice

Solve the inequality.

Explanation Slide...

When graphed in Desmos, interval can be seen.