ADV MT 2.2 Solving Equations Algebraically
Write in standard form and solve.
Standard Form of a Quadratic Equation
ax + bx + c = 0, where a, b, c ∈ R, a ≠0
x(x+2) = 7x+6
2
Solve.
2x2 + 1 =17
4x2 - 6 = 42
3(x - 2)2 = 21
1/5(x + 4)2 = 6
x2 - 9 = 12 - 2x2
(x - 3)2 = (x + 1)2
x2 -2x = 9 - 3(x-1)
(x - 1)2 = 2x
2
Nov 20­8:39 AM
Nov 30­8:15 AM
Completing the square of x2 + 8x + 9
Completing the Square
First, recall the perfect square trinomial
x2 +6x + 9
x2 -10x + 25
Find the value of c that would make the trinomial a perfect
square
x2 + 12x + c
x2 - 18x + c
x2 - 11x + c
x2 + 5x + c
x2 +8x +
= -9 +
x2 + 8x +
= -9 +
(x + 4)2
=7
√(x+4)2
= ±√7
x + 4 = ±√7
x = -4 ± √7
Nov 20­8:46 AM
x2 -6x -15 = 0
x2 -6x +
(x + 3)2
Nov 20­8:48 AM
x2 +4x - 12= 0
= 15 +
=
1. Get constant term on other side of =
2. Determine # to put in box
3. Write trinomial in factored form
4. Use algebra to solve for x
Quadratic Formula
Let a, b, c ∈ R ∋ a ≠0. The solutions of the quadratic equation:
ax + bx + c = 0 are:
x = -b ± √b - 4ac
2a
2
2
3x2 + 8x - 35 = 0
2x2 -12x +13 = -5
x2 +3x + 3 = 0
Nov 20­8:48 AM
Nov 20­8:51 AM
1
Solve using the quadratic formula
-2x2 = -2x + 3
FACTORING
x2 + bx + c
find factors of c that add up to b
both terms are same sign--use sign of b
x2 +8x + 15
x2 - 7x + 12
2x2 + x = x2 - 2x + 4
x2 - 15x + 14
x2 + 14x + 48
x2 = 2x - 5
Nov 20­8:52 AM
Nov 20­8:53 AM
x2 +bx - c
ax +bx + c
find factors of c that add up to b
factors are opposite signs
MARFF
2
x2 -x - 6
x2 + 12x - 64
Multiply a * c
Add factors of product to get b
Rewrite middle term using 2 factors
Factor
Factor
5x +17x+14
2
x2 +2x -120
4x -4x-3
2
x2 - 6x - 72
8x2 - 18x + 9
Nov 20­8:54 AM
3x2 +32x -11
Nov 20­8:54 AM
DISCRIMINANT = b2 - 4ac
• expression inside radical of quadratic formula
• determines the type & number of solutions an equation has
SOLVE BY FACTORING.
x2 - x - 12 = 0
x2 - 9x + 14 = 0
x2 - 12x + 35 = 0
x2 + 16x - 36 = 0
In a quadratic equation, ax2 + bx + c = 0,
• if b2 - 4ac > 0, there are 2 REAL SOLUTIONS
if b2 - 4ac = 0, there is 1 REAL SOLUTION
if b2 - 4ac < 0, there are NO REAL SOLUTIONS
EVALUATE THE DETERMINANT TO DETERMINE HOW
MANY REAL NUMBER SOLUTIONS
4x2 +14x + 6 = 0
2x2 - 5x - 3 = 0
9x2 + 6x + 1 = 0
9x2 + 6x - 4 = 0
9x2 + 6x + 5 = 0
Dec 12­9:55 AM
Nov 20­8:54 AM
2
WORD PROBLEMS
If a soccer field is 25 yards longer than it is wide and its
total area is 7200 ft2, what are the dimensions of the field?
Two planes leave LA at the same time, plane A heading due east &
plane B heading due south. If both fly at constant speeds with
plane A flying 50 mph faster than B, after flying for 1.5 hours the
planes are 870 miles apart. How fast are the planes flying?
LA
db = rt
LA
db = r(1.5)
Nov 20­8:57 AM
d
da = (r+50)1.5
A free falling object takes t seconds to fall d feet. t = 1/4√d
If a relief plane drops food at 1200 ft, how long does it take
for the food to reach the ground?
d = rt
da = (r+50)t
plug in t = 1.5 & d = 870
Use Pythagorean Theorem
d=870
Nov 21­10:02 AM
projectile motion
Suppose an object is launched vertically from a point so feet above
the ground with an initial velocity position of s (in feet) of the
object t seconds after it is launched, is s = -16t2 + vot + so
in a free fall, the initial velocity is zero
A pitching machine throws a baseball from ground level with initial
velocity of 55 ft/sec.
Does the ball reach a height of 52 ft?
How many seconds after the ball is propelled is it 42 ft in air?
Nov 13­2:24 PM
3