6.5 – solving equations with quadratic techniques

6.5 – Solving Equations with Quadratic Techniques

Quadratic equations are in the form:

ax2 + bx + c,

Quadratic equations are in the form:

ax2 + bx + c, where a, b, & c are integers

Quadratic equations are in the form:

ax2 + bx + c, where a, b, & c are integers

exs.

Quadratic equations are in the form:

ax2 + bx + c, where a, b, & c are integers

exs.

• x2 + 5x + 2

Quadratic equations are in the form:

ax2 + bx + c, where a, b, & c are integers

exs.

• x2 + 5x + 2

• 2x2 – 18x + 13

Quadratic equations are in the form:

ax2 + bx + c, where a, b, & c are integers

exs.

• x2 + 5x + 2

• 2x2 – 18x + 13

• x2 – 9

Quadratic equations are in the form:

ax2 + bx + c, where a, b, & c are integers

exs.

• x2 + 5x + 2

• 2x2 – 18x + 13

• x2 – 9

x2 + 0x – 9

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs.

• x2 + 5x + 2• 2x2 – 18x + 13• x2 – 9

x2 + 0x – 9• 2x2 + 8x

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs.

• x2 + 5x + 2• 2x2 – 18x + 13• x2 – 9

x2 + 0x – 9• 2x2 + 8x

2x2 + 8x + 0

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs.

• x2 + 5x + 2• 2x2 – 18x + 13• x2 – 9

x2 + 0x – 9• 2x2 + 8x

2x2 + 8x + 0NOTE: Must have the “x2” term to be a quadratic!

Ex. 1 Write each expression in quadratic form, if possible.

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36

(x2)2

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36

(x2)2 + 13(x2)

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36

(x2)2 + 13(x2) + 36

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36

(x2)2 + 13(x2) + 36

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36

(x2)2 + 13(x2) + 36

b. 16x6 – 625

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36

(x2)2 + 13(x2) + 36

b. 16x6 – 625

(4x3)2

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36

(x2)2 + 13(x2) + 36

b. 16x6 – 625

(4x3)2 – 625

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 – 625 (4x3)2 – 625

c. x½ – 9x¼ + 16

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 – 625 (4x3)2 – 625

c. x½ – 9x¼ + 16 (x¼)2

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 – 625 (4x3)2 – 625

c. x½ – 9x¼ + 16 (x¼)2 – 9(x¼)

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 – 625 (4x3)2 – 625

c. x½ – 9x¼ + 16 (x¼)2 – 9(x¼) + 16

Ex. 1 Write each expression in quadratic form, if possible.

a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 – 625 (4x3)2 – 625

c. x½ – 9x¼ + 16 (x¼)2 – 9(x¼) + 16

Ex. 2 Solve each equation.a. x4 = 16

Ex. 2 Solve each equation.a. x4 = 16

-16 -16

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 )(x2 ) = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x )(x ) = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0

x – 2 = 0 OR x + 2 = 0

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0

x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0

x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0

x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0

x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x2 + 4 = 0

x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x2 + 4 = 0

x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2

x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x2 + 4 = 0

x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4

x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x2 + 4 = 0

x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4

x = 2 OR x = -2 OR x = ±2i

Ex. 2 Solve each equation.a. x4 = 16

-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

( )( ) = 0(x2 – 4)(x2 + 4) = 0

x2 – 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x2 + 4 = 0

x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4

x = 2 OR x = -2 OR x = ±2i

b. x4 + 11x2 + 18 = 0

b. x4 + 11x2 + 18 = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 )(x2 ) = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

x2 + 9 = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

x2 + 9 = 0 OR x2 + 2 = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

x2 + 9 = 0 OR x2 + 2 = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

x2 + 9 = 0 OR x2 + 2 = 0

(x + )(x + ) = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0

x2 + 9 = 0

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0

x2 + 9 = 0

-9 -9

b. x4 + 11x2 + 18 = 0

(x2)2 + 11(x2) + 18 = 0

( )( ) = 0

(x2 + 9)(x2 + 2) = 0

x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0

x2 + 9 = 0

-9 -9

x2 = -9

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 -9 -9

x2 = -9 √x2 = √-9

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 -9 -9

x2 = -9 √x2 = √-9

x = ±3i

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 -9 -9

x2 = -9 √x2 = √-9

x = ±3i

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 (x + )(x + ) = 0 x2 + 9 = 0 -9 -9

x2 = -9 √x2 = √-9

x = ±3i

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 OR x2 + 2 = 0 -9 -9

x2 = -9 √x2 = √-9

x = ±3i

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 OR x2 + 2 = 0 -9 -9 -2 -2

x2 = -9 √x2 = √-9

x = ±3i

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 OR x2 + 2 = 0 -9 -9 -2 -2

x2 = -9 OR x2 = -2 √x2 = √-9

x = ±3i

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 OR x2 + 2 = 0 -9 -9 -2 -2

x2 = -9 OR x2 = -2 √x2 = √-9 OR √x2 = √-2

x = ±3i

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 OR x2 + 2 = 0 -9 -9 -2 -2

x2 = -9 OR x2 = -2 √x2 = √-9 OR √x2 = √-2

x = ±3i OR x = ±i√2

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0( )( ) = 0

(x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

( )( ) = 0 ( )( ) = 0 x2 + 9 = 0 OR x2 + 2 = 0 -9 -9 -2 -2

x2 = -9 OR x2 = -2 √x2 = √-9 OR √x2 = √-2

x = ±3i OR x = ±i√2

c. x½ – 6x¼ – 16 = 0

c. x½ – 6x¼ – 16 = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

( )( ) = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

( )( ) = 0

(x¼ )(x¼ ) = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

( )( ) = 0

(x¼ – 8)(x¼ + 2) = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

( )( ) = 0

(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

( )( ) = 0

(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

( )( ) = 0

(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0

( )( ) = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

( )( ) = 0

(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0

( )( ) = 0

x¼ – 8 = 0

c. x½ – 6x¼ – 16 = 0

(x¼)2 – 6(x¼) – 16 = 0

( )( ) = 0

(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0

( )( ) = 0

x¼ – 8 = 0

+8 +8

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0

x¼ – 8 = 0 +8 +8 x¼ = 8

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0

x¼ – 8 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0

x¼ – 8 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4 x = 4096

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0

x¼ – 8 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4 x = 4096

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0

x¼ – 8 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4 x = 4096

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4 x = 4096

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 (x¼)4 = (8)4 x = 4096

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 OR x¼ = -2 (x¼)4 = (8)4 x = 4096

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 OR x¼ = -2 (x¼)4 = (8)4 OR (x¼)4 = (-2)4 x = 4096

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 OR x¼ = -2 (x¼)4 = (8)4 OR (x¼)4 = (-2)4 x = 4096 OR x = 16

c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0

x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 OR x¼ = -2 (x¼)4 = (8)4 OR (x¼)4 = (-2)4 x = 4096 OR x = 16