Math Review
Terminology
Discriminant: The discriminant shows whither there are real roots or not. If the discriminant is a
negative, it means that there are no real roots. If the discriminant is zero, there are two equal roots.
Otherwise there are two real roots.
Zeros: Zeroes are the x-intercepts on a parabola. (X-intercepts are when the parabola crosses the x-axis.)
If there are no zeros this means that the parabola will never cross the x-axis.
Roots: Two distinct roots means when doing the quadratic formula the number under the square root
sign is positive. Two equal roots means the number under the square root sign (in the quadratic
formula) is zero. No real roots means the number under the square root sign (in the quadratic formula)
is a negative sign.
Formulas:
The Quadtractic Formula
√
The Quadratic Formula gives you the roots, of an equation
The “New Best Friend”
This will give you h, which is the x-value of the vertex.
Forms:
Vertex Form: y=a(x-h) +k
Standard Form: y=ax²+bx+c
Factored Form: a(x-xˌ) (x-x2)
Examples With step by step:
Using the quadratic formula to find x-intercepts
Step 1. Identify a,b,c
4x2+2x-5=0
A=4
B=2
C=-5
√
Step 2. Write the formula you need to use. (Quadratic formula)
√
Step 3. Fill in with a,b,c.
√
Step 4. Complete the exponents.
√
Step 5. Do all the multiplication
√
Step 6. Complete the addition or subtraction for the discriminant.
Step 7. Find the square root of the discriminant.
Step 8. Do an equation for both addition and subtraction.
Step 9. Simplify the numerator of both fractions.
X=o.9
x= -1.4
Step 10. Solve the fraction to find your roots.
Therefore the approximate roots for this equation are 0.9 and -1.4.
using “Your new Best friend”
Finding The x-value of the vertex.
4x2+2x-5=0
Step 1. Identify the a,b,c values.
A=4
B=2
c= -5
Step 2. Identify the formula you need. (your new best friend)
h=
Step3. Input you’re a,b,c values into the formula.
h=
Step 4. Solve the fraction to find your x-value of the vertex.
h=-0.25
Therefore the x-value of the vertex is -0.25.
Application. Try it yourself.
Application question 1.
A bridge in Utah is represented by the equation -0.04x2+3.28x=0.
Using this equation find the horizontal distance of the bridge in meters.
Find the x-value of the vertex.
Find the y-value of the vertex.
What is the discriminant?
Application question 2.
A rectangular pool that is 4 meters by 6 meters needs a patio that has the same width all the way
around. If the area of the patio is equal to the area of the pool, determine the width of the patio.
Answers to the Try it yourself application questions
Answers To Question 1
Finding the roots
-0.04x2+3.28x=0
√
√
√
√
X=0
x=82
Therefore the roots are 0,82. Therefore the distance across the bridge, is 82 meters.
Also by using the quadratic formula, we can find the discriminant the discriminant is 10.7584, this means
that There will be two positive roots.
Finding the x-value of the vertex.
h= 41
The x-value of the vertex is 41.
Finding the y-value of the vertex.
-0.04(41)2+3.28(41)=y
-0.04(1681)+3.28(41)=y
-67.24+134.48=y
67.24=y
The y-value of the vertex is 67.24
Answer to question 2.
6 Meters
4 Meters
A=24
24=(4+2x)(6+2x)
24=24+8x+12x+4x2
24=4x2+20+24
0=4(x2+5x-6)
0=4(x+6)(x-1)
Roots: x=-6, x=1
The width of the patio can’t be negative, therefore it is 1 meter.
By: Elyse Mann, Robbyn Wilkinson, Mary Sission