Quadratic Equations
How many real solutions does the following equation have?
x² - 49 = 0
Two real solutions!
b² - 4ac = 0² - 4(1)(-49) = 196 > 0
Simplify: √(100v³)
10v√(v)
What is the vertical asymptote for the following function? Give the equation.
y = 2/x
x = 0
Evaluate f(3).
f(x) =
x - 1 if x ≤ 0
2x if x > 0
f(3) = 2(3) = 6
What is the value of the discriminant for the following quadratic?
3x² + 5x - 18 = 0
b² - 4ac = 5² -4(3)(-18) = 241
How many real solutions does the following equation have?
x² + 3x + 4 = 0
No real solutions!
b² - 4ac = 3² - 4(1)(4) = 9 - 16 = -7 < 0
Simplify:
-3√18 + 3√8 - √24
-3(3√2) + 3(2√2) - 2√6
-9√2 + 6√2 - 2√6
-3√2 - 2√6
What is the horizontal asymptote for the following function? Give the equation.
y = 1/x
y = 0
Evaluate f(-1)
f(x):=
2x² if x ≤ 0
3x if x > 0
f(-1) = 2(-1)² = 2(1) = 2
What is the value of the discriminant for the quadratic?
-6x² - 3x = -4
b² - 4ac = (-3)² -4(-6)(4) = 9 - 96 = 105
Solve the following:
5x + 3x² = 8
x = -8/3, 1
Simplify:
1 / (8 - 2√5)
(4 + √5) / 22
What is the vertical asymptote for the following function? Give the equation.
y = (2x) / (x² - 9)
x = 3, x = -3
x² - 9 = 0
x² = 9
x = ±3
f(x):=
-x. if x ≥ 0
x. if x < 0
f(-121) = -121
Simplify:
2√32 + 3√8 - 4√18
2(4√2) + 3(2√2) - 4(3√2) = 2√2
Solve the following:
3x² - 6x - 12 = 0
x = 1 +√5, 1 - √5
Solve:
√(x - 4) = 3
x - 4 = 9
x = 13
What is the horizontal asymptote for the following function? Give the equation.
y = 2/(x+1) + 3
y = 3
Evaluate g(1).
g(x) =
x + 8. if x ≤ 1
3 - x. if x > 1
g(1) = 1 + 8 = 9
Simplify:
√(2x³y)√(12xy)
√(24x4y²) = 2x²y√6
Solve the following:
8x² + 4x - 16 = -x²
x = (-2 ± 2√37)/9
Solve:
√(30 - x) = x
30 - x = x²
0 = x² + x - 30
0 = (x - 5)(x + 6)
x = 5, -6 —> -6 doesn’t work!
Solution: x = 5
What is the horizontal asymptote for the following function? Give the equation.
(x³ - 1) / (x² - 9)
Degree of numerator > degree of denominator
There is no horizontal asymptote!
Evaluate f(4)
f(x)=
x² if x ≤ -4
2x³ + 1 if -4 < x ≤ 4
3x - 2. if x > 4
f(4) = 2(4)³ + 1 = 2(64) + 1 = 128 + 1 = 129
Simplify:
(√2 - 5)²
(√2 - 5)(√2 - 5) = 2 - 5√2 - 5√2 + 25 = 27 - 10√2