Solving Quadratic Equations By The Formula In Graphic Form Worksheet With Answers By Nghi H Nguyen (Page 3 of 3) in pdf

Solving Quadratic Equations By The Formula In Graphic Form Worksheet With Answers By Nghi H Nguyen Page 3

Examples of solving quadratic equations by the quadratic formula in graphic form.

Example 1. Solve:

4x^2 – 12x + 9 = 0.

Solution. First, find d by the relation (2):

d^2 = b^2 – 4ac = 144 – 144 = 0  d = 0

The equation has double root at x = -b/2a = 12/8 = 3/2.

Example 2. Solve:

2x^2 – 3x + 7 = 0

Solution. First find d^2 by the relation (2).

d^2 = 9 – 56 = -47.

d is imaginary, there are no real roots.

Example 3. Solve:

3x^2 + 16x - 12 = 0.

Solution. d^2 = 256 + 144 = 400 = (20)^2

Next, find the 2 real roots by the formula (1).

x1 = -16/6 + 20/6 = 4/6 = 2/3

x2 = -16/6 – 20/6 = -36/6 = -6.

Example 4. Solve:

7x^2 + 18x – 25 = 0.

Solution. d^2 = 324 + 700 = 1024 = (32)^2

x1 = -18/14 + 32/14 = 14/14 = 1

x2 = -18/14 – 32/14 = -50/14 = -25/7

Example 5. Solve :

2x^2 + 12x + 17 = 0

Solution. d^2 = 144 – 136 = 8  d = 2.83

x1 = -12/4 + 2.83/4 = -9.17/4 = -2.29

x2 = -12/4 – 2.83/4 = -14.83/4 = -3.70

Example 6. Solve:

5x^2 – 10x – 3 = 0.

Solution. d^2 = 100 + 60 = 160  d = 12.65

x1 = 10/10 + 12.65/10 = 22.65/10 = 2.26

x2 = 10/10 – 12.65/10 = -2.65 /10 = -0.26

NOTE. When the given quadratic equation can be factored, its 2 real roots are usually in the

form of two fractions. The quantity d^2 should be a perfect square and d should be a whole

number. So, students are advised to proceed solving by the formula in graphic form, mentally

or by using calculators, through 2 steps. First step, find d by the relation (2). Second step,

algebraically calculate the 2 real roots by the formula (1). In case d is a whole number, make

sure that the 2 real roots (answers) be in the form of 2 fractions and not in decimals.

(This article was written by Nghi H. Nguyen, co-author of the new Diagonal Sum Method for

solving quadratic equations)

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