Equations in Quadratic Form
Equations in Quadratic Form
Equations in Quadratic Form There are many methods for solving quadratic equations.
ax bx c 0 2
If an equation has the pattern of a quadratic equation, we can use one of these methods to solve it.
Solving Equations in Quadratic Form Step 1: Arrange the equation in quadratic form.
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0
Let t = (x2 + 2x)
Solving Equations in Quadratic Form Step 2: Substitute t into the equation.
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0
Let t = (x2 + 2x) t2–2t–3=0
Solving Equations in Quadratic Form Step 3: Choose the best method for solving the quadratic equation and then solve for t.
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0
Let t = (x2 + 2x) t2–2t–3=0
In this case, let’s factor!
(t + 1) (t – 3) = 0 (t – 3) = 0 t=3
(t + 1) = 0 t=–1
Solving Equations in Quadratic Form Step 4: Solve for x.
t=3 t = (x2 + 2x) 3 = x2 + 2x 0 = x2 + 2x –3 0 = (x + 3)(x – 1) 0 = (x + 3) 0 = (x – 1) x = –3 x=1
t=–1 t = (x2 + 2x) –1 = x2 + 2x 0 = x2 + 2x +1 0 = (x + 1)(x + 1) 0 = (x + 1) x = –1
Solving Equations in Quadratic Form Checking Solutions
x = –3
x=1
x = –1
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0 ((–3)2 + 2(–3))2 – 2 ((–3)2 + 2(–3)) – 3 = 0 (9 + (–6))2 – 2 (9 + (–6)) – 3 = 0 (3)2 – 2 (3) – 3 = 0 9–6–3=0 0=0
Solving Equations in Quadratic Form Checking Solutions
x = –3
x=1
x = –1
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0 ((1)2 + 2(1))2 – 2 ((1)2 + 2(1)) – 3 = 0 (1 + 2)2 – 2 (1 + 2) – 3 = 0 (3)2 – 2 (3) – 3 = 0 9–6–3=0 0=0
Solving Equations in Quadratic Form Checking Solutions
x = –3
x=1
x = –1
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0 ((–1)2 + 2(–1))2 – 2 ((–1)2 + 2(–1)) – 3 = 0 (1 + (–2))2 – 2 (1 + (–2)) – 3 = 0 (–1)2 – 2 (–1) – 3 = 0 1+2–3=0 0=0
Solving Equations in Quadratic Form Checking Solutions
x = –3
x=1
x = –1
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0 ((–1)2 + 2(–1))2 – 2 ((–1)2 + 2(–1)) – 3 = 0 (1 + (–2))2 – 2 (1 + (–2)) – 3 = 0 (–1)2 – 2 (–1) – 3 = 0 1+2–3=0 0=0
Solve the following equation:
𝟐𝒙𝟒 + 𝟏𝟏𝒙𝟐 – 6 = 0
Solve the following equation:
𝟐𝒙𝟒 + 𝟏𝟏𝒙𝟐 – 6 = 0
Solve the following equation:
𝟐𝒙𝟒 + 𝟏𝟏𝒙𝟐 – 6 = 0 Checking Solutions
Simplify following expression: Solve thethe following equation:
𝑥 + 𝑥 = 20 a. b. c. d.
x = –4, x = –5 x = 4, x = 5 x = –16, x = –25 x = 16, x = 25
Solve the following equation:
𝑥 + 𝑥 = 20
Solve the following equation:
𝑥 + 𝑥 = 20
Solve the following equation:
𝑥 + 𝑥 = 20 Checking Solutions
Equations in Quadratic Form There are many methods for solving quadratic equations.
ax bx c 0 2
If an equation has the pattern of a quadratic equation, we can use one of these methods to solve it.
Solving Equations in Quadratic Form Step 1: Arrange the equation in quadratic form.
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0
Let t = (x2 + 2x)
Solving Equations in Quadratic Form Step 2: Substitute t into the equation.
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0
Let t = (x2 + 2x) t2–2t–3=0
Solving Equations in Quadratic Form Step 3: Choose the best method for solving the quadratic equation and then solve for t.
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0
Let t = (x2 + 2x) t2–2t–3=0
In this case, let’s factor!
(t + 1) (t – 3) = 0 (t – 3) = 0 t=3
(t + 1) = 0 t=–1
Solving Equations in Quadratic Form Step 4: Solve for x.
t=3 t = (x2 + 2x) 3 = x2 + 2x 0 = x2 + 2x –3 0 = (x + 3)(x – 1) 0 = (x + 3) 0 = (x – 1) x = –3 x=1
t=–1 t = (x2 + 2x) –1 = x2 + 2x 0 = x2 + 2x +1 0 = (x + 1)(x + 1) 0 = (x + 1) x = –1
Solving Equations in Quadratic Form Checking Solutions
x = –3
x=1
x = –1
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0 ((–3)2 + 2(–3))2 – 2 ((–3)2 + 2(–3)) – 3 = 0 (9 + (–6))2 – 2 (9 + (–6)) – 3 = 0 (3)2 – 2 (3) – 3 = 0 9–6–3=0 0=0
Solving Equations in Quadratic Form Checking Solutions
x = –3
x=1
x = –1
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0 ((1)2 + 2(1))2 – 2 ((1)2 + 2(1)) – 3 = 0 (1 + 2)2 – 2 (1 + 2) – 3 = 0 (3)2 – 2 (3) – 3 = 0 9–6–3=0 0=0
Solving Equations in Quadratic Form Checking Solutions
x = –3
x=1
x = –1
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0 ((–1)2 + 2(–1))2 – 2 ((–1)2 + 2(–1)) – 3 = 0 (1 + (–2))2 – 2 (1 + (–2)) – 3 = 0 (–1)2 – 2 (–1) – 3 = 0 1+2–3=0 0=0
Solving Equations in Quadratic Form Checking Solutions
x = –3
x=1
x = –1
(x2 + 2x)2 – 2 (x2 + 2x) – 3 = 0 ((–1)2 + 2(–1))2 – 2 ((–1)2 + 2(–1)) – 3 = 0 (1 + (–2))2 – 2 (1 + (–2)) – 3 = 0 (–1)2 – 2 (–1) – 3 = 0 1+2–3=0 0=0
Solve the following equation:
𝟐𝒙𝟒 + 𝟏𝟏𝒙𝟐 – 6 = 0
Solve the following equation:
𝟐𝒙𝟒 + 𝟏𝟏𝒙𝟐 – 6 = 0
Solve the following equation:
𝟐𝒙𝟒 + 𝟏𝟏𝒙𝟐 – 6 = 0 Checking Solutions
Simplify following expression: Solve thethe following equation:
𝑥 + 𝑥 = 20 a. b. c. d.
x = –4, x = –5 x = 4, x = 5 x = –16, x = –25 x = 16, x = 25
Solve the following equation:
𝑥 + 𝑥 = 20
Solve the following equation:
𝑥 + 𝑥 = 20
Solve the following equation:
𝑥 + 𝑥 = 20 Checking Solutions
