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10.1-10.2
MATH 60/GRACEY
When you are done with your 10.1 homework you should be able to…
π
π
π
π
Solve quadratic equations using the Square Root Property
Complete the square in one variable
Solve quadratic equations by completing the square
Solve problems using the Pythagorean Theorem
WARM-UP:
1. Factor.
3 x 2 − 16 x + 21
2. Solve.
x2 − 3 = 2 x
2
3. Graph f ( x ) = x .
Domain:
Range:
10.1-10.2
MATH 60/GRACEY
The Square Root Property
2
If x = p , then x =
p or x = − p .
1. Solve. Assume that we are working in the complex numbers.
a. x 2 = 20
b. m 2 + 121 = 0
Solving a Quadratic Equation Using the Square Root Property
Step 1: Isolate the expression containing the squared term.
Step 2: Use the Square Root Property. Don’t forget the ± symbol.
Step 3: Isolate the variable, if necessary.
Step 4: Check your solution.
2. Solve.
a. 7 r 2 = 112
b. 8 − z 2 = 108
c.
( a − 2)
2
+ 12 = 0
10.1-10.2
MATH 60/GRACEY
Obtaining a Perfect Square Trinomial
Identify the coefficient of the first-degree term. Multiply this coefficient by
1
and then square the result. That is, determine the value of b in x 2 + bx + c and
2
2
1 
compute  b  .
2 
3. Determine the number that must be added to the expression to make it a
perfect square trinomial. Then factor the expression.
a. w2 + 3w
b. x 2 + 18 x
Solving a Quadratic Equation by Completing the Square
Step 1: Rewrite x 2 + bx + c = 0 as x 2 + bx = −c by subtracting the constant from
both sides of the equation.
Step 2: Complete the square in the expression x 2 + bx by making it a perfect
square trinomial. Don’t forget, whatever you add to the left side of the equation
must also be added to the right side.
Step 3: Factor the perfect square trinomial on the left side of the equation.
Step 4: Solve the equation using the Square Root Property.
Step 5: Verify your solutions.
4. Solve by completing the square.
a. n 2 + 8n + 4 = 0
b. x 2 + 5 x + 1 = 0
10.1-10.2
MATH 60/GRACEY
The Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum
of the squares of the lengths of the legs.
c2 = a 2 + b2
5. A fire lookout station for the National Park Service is 160 feet above the
ground. Suppose that a park ranger climbs to the top of the lookout station
to look for forest fires. How far can the park ranger see? Assume the
radius of the Earth 3960 miles.
10.1-10.2
MATH 60/GRACEY
When you are done with your 10.2 homework you should be able to…
π Solve quadratic equations using the quadratic formula
π Use the discriminant to determine the nature of solutions in a quadratic
equation
π Model and solve problems involving quadratic equations
The Quadratic Formula
The solution(s) to the quadratic equation
ax 2 + bx + c = 0, a ≠ 0 , are given by
the quadratic formula
−b ± b 2 − 4ac
x=
2a
.
Solving a Quadratic Equation Using the Quadratic Formula
Step 1: Write the equation in standard form
ax 2 + bx + c = 0
and identify the
values of a, b, and c.
Step 2: Substitute the values of a, b, and c into the quadratic formula.
Step 3: Simplify the expression found in step 2.
Step 4: Verify your solution(s).
10.1-10.2
MATH 60/GRACEY
6. Solve.
a.
y 2 − 2 = −4 y
b.
16k +
c.
m2 + m + 2 = 0
9
= −24
k
10.1-10.2
MATH 60/GRACEY
The Discriminant and the Nature of the Solution of a Quadratic Equation
For a quadratic equation
ax 2 + bx + c = 0 , the discriminant b 2 − 4ac
can be used
to describe the nature of the solution as shown:
Discriminant
Number of Solutions
Type of Solution
Positive and a
perfect square
Positive and not a
perfect square
Zero
Negative
7. For each quadratic equation, determine the discriminant. Use the value of
the discriminant to determine whether the quadratic equation has two
unequal rational solutions, two irrational solutions, one repeated real solution,
or two complex solutions that are not real.
a.
x2 − 5x + 3 = 0
b.
4n 2 + 12n + 9 = 0
c.
4 w 2 − 4 w = −5
10.1-10.2
MATH 60/GRACEY
8. The revenue R received by a company selling x specialty T-shirts per week is
given by the function
R ( x ) = −0.005 x 2 + 30 x .
a. How many T-shirts must be
sold in order for revenue
to be $40,000 per week?
b. How many T-shirts must be
sold in order for revenue
to be $33,750 per week?