Download Quadratic Equations Assignment_2

Math 20-1
Quadratic Functions and Equations
Name: ____________________
Solving Quadratic Equations
Specific Outcome
R&F 5: Solve problems that involve quadratic
equations.
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Questions
all
Total
47
Solving a quadratic equation means finding what value of the variable makes both sides of
the equation equal.
Solutions have the form x  ______ .
A quadratic equation can have two, one, or no solutions.
There are three ways to solve a quadratic equation algebraically:
1. Factoring
2. Completing the Square
3. Quadratic Formula
For all methods of solving, you must first move all the terms to one side so that the
equation = 0.
Method 1: Solving by Factoring
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- GCF
- Difference of squares
- Trinomial factoring (PSF / Grid)
Then set each factor = 0 and solve
1. Find the roots of the following equations by factoring. (2 marks each)
a) x 2  6 x  27
b) 4h 2  49  0
c)
1 2
d  3d
4
d)
n  3n  4  6
Math 20-1
Quadratic Functions and Equations
e) x 2 
7
x20
2
f) 9 x 2  7 x  6  5 x  2
2. A field is in the shape of a right triangle. The fence around the perimeter of the field is 40 m
in total. The length of the hypotenuse is 17 m. Find the length of the other two sides of the
right triangle by writing and solving a quadratic equation by factoring. (3 marks)
Math 20-1
Quadratic Functions and Equations
Method 2: Solving by Completing the Square
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Best if the equation is already in vertex form
Don’t forget the  square root to get two solutions!
3. Find the roots of the following quadratic equations by completing the square. Write your
answers in exact value, if necessary.
1
2
a)   x  8  1  0 (2 marks)
4
b) 5 x 2  10 x  2  0 (3 marks)
Math 20-1
Quadratic Functions and Equations
Method 3: Solving by Quadratic Formula
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 b  b 2  4ac
Quadratic formula: x 
2a
Rounded to the nearest tenth, hundredth, etc: two answers
Exact value – simplify the radical (and the fraction, if possible)
Square root of a negative means there are no real roots.
4. Find the roots of the following quadratic equations using the quadratic formula. Write your
answers as integers or fractions if possible, or round to the nearest hundredth, if necessary.
(2 marks each)
a) 4 x 2  8 x  3  0
b) 3 x 2  4 x  7
5. Solve the following equations by using the quadratic formula. Write all answers in simplest
exact form. (3 marks each)
a) 5n 2  7  11n
b) 1  x 2  4 x
Math 20-1
Quadratic Functions and Equations
6. The surface area of a cone is 300 cm2 and its slant height is 15 cm. Use the surface area
formula A  r 2  rs to determine the radius of the cone to the nearest hundredth of a
centimetre. (3 marks)
7. A rectangle has a length that is 7 ft longer than its width. The numerical value of the area of
the rectangle (in ft2) is equal to the numerical value of the perimeter of the rectangle (in ft).
Find the dimensions of the rectangle to the nearest hundredth, and its area and perimeter to
the nearest tenth. (4 marks)
Math 20-1
Quadratic Functions and Equations
8. Find the value of the discriminant for each quadratic equation. Then say how many roots the
equation has. (2 marks each)
a) 5 x 2  4 x  12  0
b)  36 x 2  12 x  1  0
c) 8 x  2 x 2  3
d) x 2  10  14 x  10
9. For what values of c does 18 x  3x 2  c have no real roots? (2 marks)