Remediation 2.2 CFA Quadratic Formula

The standard form of a quadratic equation is ax²+bx+c

What are the values for a, b and c in the following quadratic equations?

3x²+5x-1=0 2x²-7x=10
a=3, b=5, c=-1 a=2, b=-7, c=-10 (set = to 0)

Multiple Choice

Determine the values of
a, b, and c for
the quadratic equation: 
4x² – 8x = 3
*remember right side should =0

• a = 4, b = -8, c = 3
• a = 4, b =-8, c =-3
• a = 4, b = 8, c = 3
• a = 4, b = 8, c = -3

Multiple Choice

Identify a, b, and c in: x²−6x+14

• a=3,b=5,c=9

• a=1,b=-6,c=14

• a=1,b=-6,c=-14

• a=1,b=6,c=14

Solving Quadratics ~ Process for Using the Quadratic Formula

Solving Quadratics ~ Example

Multiple Choice

What are the zeros of

$$f\left(x\right)=x^2+7x-18$$  ?

• $$x=-2,\ -9$$  

• $$x=2,\ -9$$  

• $$x=-2,\ 9$$  

• $$x=2,\ 9$$  

Multiple Choice

Solve using the quadratic formula:

4x² + 4x + 1 = 0

• x = -½

• x = -2

• x = 0

• x = ½

Multiple Choice

Use the quadratic formula to find the solutions for

y = -x² - 5x + 12

• No Real Solution

The Discriminant will tell you the number of solutions you have.

Discriminant = b² - 4ac

If the discriminant is:

• positive , then you have 2 real solutions

• negative, then you have 2 complex solutions

• zero - then you have one real solution (double root)

Multiple Choice

If the discriminant of the quadratic equation is equal to 0, how many real solutions are there?

• None

• One

• Two

• Infinite

Multiple Choice

If the discriminant of the quadratic equation is equal to 4, how many real solutions are there?

• None

• One

• Two

• Infinite

Multiple Choice

How many solutions does this equation have?

• 2 Real Solutions

• 1 Real Solution

• 2 Complex Solutions

Multiple Choice

What is the discriminant of $$4x^2+8x+6=0$$ AND what type of solutions does it have?

• -32; 2 complex roots

• 32; 2 irrational roots

• 160; 2 irrational roots

• 16; 2 rational roots

Multiple Choice

$$5x^2=10x-5$$  

What is the discriminant of the equation AND what type of solutions does it have?

• 0; 1 rational root

• 0; 2 irrational roots

• 200; 2 irrational roots

• 200; 2 complex roots

Multiple Choice

What is the discriminate of f(x) = 3x² - 5x + 8
and how many solutions does the function have?

• discriminate = -71 and two complex solutions

• discriminate = -121 and has two complex solutions

• the discriminate = - 71 and has two real solutions

• the disctiminate = -121 and has two real solutions

Applications

When solving application problems dealing with quadratics, set the equation equal to 0 and use the quadratic formula. Make sure to check your answers. Choose the answer that makes sense!
If you are finding how many seconds it will take for a ball to reach the ground, you will not choose the negative solution!

Multiple Choice

The function f(t) = -5t²+20t + 60 models the approximate height of an object t seconds after it is launched. How many seconds does it take the object to hit the ground? 

• 4 seconds
• -2 seconds
• 6 seconds

• 9 seconds

Multiple Choice

A frog is sitting on the ground when he is scared by a big dog. His movement is modeled by the equation h = -6t² + 6t where h is the frog's height at any given time, t . How many seconds until the frog is back on the ground?

• 0.5 seconds

• 1 second

• 1.5 seconds

• 2 seconds

• 3 seconds

Multiple Choice

The function f(t) = -5t²+20t + 60 models the approximate height of an object t seconds after it is launched. How many seconds does it take the object to hit the ground?

• 4 seconds

• 2 seconds

• 6 seconds

• 9 seconds

Complex Numbers

Learning Target: I can simplify expressions with complex numbers.

Multiple Choice

Solve using the quadratic formula.

$$r^2-2r+4=0$$  

• $$\left\{1\right\}$$  

• $$\left\{1+i\sqrt[]{3},\ 1-i\sqrt[]{3}\right\}$$  

• $$\left\{1.618,\ -0.618\right\}$$  

• $$\left\{\frac{1+i\sqrt[]{3}}{2},\ \frac{1-i\sqrt[]{3}}{2}\right\}$$  

Multiple Choice

Solve using the quadratic formula. $$f\left(x\right)=2x^2-4x+7$$  

• $$\frac{2\pm2i\sqrt{10}}{2}$$  

• $$\frac{4\pm\sqrt{40}}{4}$$  

• $$\frac{2\pm i\sqrt{10}}{2}$$  

• $$\frac{2\pm2i\sqrt{40}}{4}$$  

Adding Complex Numbers

• When adding complex numbers

  -Add real parts together

  -Add imaginary parts together

Multiple Choice

Find the sum.

(5-2i) + (-7+8i)

• -2+6i

• 12+6i

• -35-16i²

• -35 -16i

Subtracting Complex Numbers

• Distribute the negative

• Add real parts together

  Add imaginary parts together

Multiple Choice

Simplify:

(10+ 15i) - (48 - 30i)

• 58 - 45i

• 58 - 15i

• -38 - 15i

• -38 + 45i

Multiplying Complex Numbers

• Use an area model.

• Multiply to get insides

• Add like terms

• simplify i²

• Write as a sum

Multiple Choice

Simplify:
-3i(4 + 2i)

• 6 - 12i
• 6 + 12i
• -12i - 6i²
• -12i + 6i²

Multiple Choice

Simplify:
(2 - 3i)(5 + 4i)

• 22 - 7i
• 22 + 7i
• 22 - 7i²
• 22 + 7i²

Multiple Choice

Simplify
(-6 - 2i)²

• 32
• 36-4i
• 36+24i
• 32+24i

Imaginary Numbers

Multiple Choice

Simplify

√-4

• 2i

• -2i

• 2

• 4i

Multiple Choice

Simplify

• ± 6
• ± 6i
• ± √6i
• ± √6