PROJECT 1 AND 2

Project 1 and 2

Introduction

Project 1

An interesting method for solving quadratic equations came from India. The steps are:

(a) Move the constant term to the right side of the equation.

(b) Multiply each term in the equation by four times the coefficient of the x squared term.

(c) Square the coefficient of the original x term and *** it to both sides of the equation.

(d) Take the square root of both sides.

(e)set the left side of the equation equal to the negative square root of the number on the right side of the equation and solve for x. (sqd ) is abbreviation for x squared

Lets suppose X (sqd) + 3x -10 =0

X (sqd) + 3x =10

4X (sqd) + 12x = 40

4x (sqd) + 12x + 9 = 40 + 9

4x (sqd) + 12x + 9 = 49

2x + 3 = + or -7

2x +3=7 2x+3= -7

2x=4 2x= -10

x=2 x= -5

Solve these:

(a) x (sqd) - 2x -13 = 0

(b) 4x (sqd) - 4x +3 = 0

(c) x (sqd) +12x -64 = 0

(d) 2x (sqd) - 3x -5 = 0

Solve: 2x² - 3x - 5 = 0

2x² - 3x = 5 (move constant over to the right)

16x² - 24x = 40 (multiply through by 8)

16x² - 24x + 9 = 40 + 9 (add (-3)² = 9 to both sides)

The expression on the left hand side, gives you an expression that is a perfect square.

(4x - 3)² = 49

Square both sides

4x - 3 = ±v49

4x - 3 = ±7

4x = ±7 + 3

4x = 10 or 4x = -4

x = 10/4 or x = -1

x = 5/2 or x = -1

There is another method that I would like to demonstrate.

Solve: 2x² - 3x - 5 = 0

Multiply the two outside coefficients: 2 and -5

This gives you -10.

Find two factors of -10 that sum to -3 (middle coefficient)

Those factors are 2 and -5

Using these factors re-write the middle coefficient

2x² + 2x - 5x - 5 = 0

Proceed by factoring by grouping

(2x² + 2x) + (-5x - 5) = 0

2x(x + 1) - 5(x + 1) = 0

Factor out (x + 1)

(x + 1)(2x - 5) = 0

x + 1 = 0 or 2x - 5 = 0

x = -1 or x = 5/2 Solved

Here is another technique: (completing the square)

Solve: 2x² - 3x - 5 = 0

2(x² - 3/2x) = 5

2(x² - 3/2x + ?) = 5 + 2?

= ((-3/2)/2)² = 9/16

2(x² - 3/2x + 9/16) = 5 + 2(9/16)

2(x - 3/4)² = 49/8

(x - 3/4)² = (49/8)/2

(x - 3/4)² = 49/16

Take the square root of both sides

(x - 3/4) = ±7/4

x = ±7/4 + 3/4

x = 7/4 + 3/4 or x = -7/4 + 3/4

x = 5/2 or x = -1

Yes the negative 2x changes your equation but it doesn't make it any easier or nicer to solveYou can't factor this oneYou either need to complete the square or use the quadratic formula to complete the square.

Project 2send along the quadratic formula methodLet's start withx^2-2x=13so we take the 1/2 of the ...