The Quadratic Formula by Example 

-- Sections 1.3-1.4 --

  

ax2 + bx + c = 0

Memorize the formula.  See your book for proof (page 102).

  Solve for x:   2x2 + 3x – 1 =  0        Quadratic Formula Method

Solve

Step

Check 

a = 2, b = 3, c = -1

Identify coefficients

If x  = 0.2808, then

2(0.2808)2 +3(0.2808) -1 = 0

    0.1577 + .8424 - 1 = 0
0.0001 » 1
**Rounding error

If x  = -1.7808, then

2(-1.7808)2 + 3(-1.7808) - 1 = 0

   6.3425 - 5.3424 - 1 = 0
0.0001 » 1
**Rounding error

Substitute values for a, b , and c

Simplify 

There are 2 distinct answers.

(exact)

(exact)

Approximations...
x » (-3 + 4.1231)/ 4 = 0.2808
  x » (-3 – 4.1231)/ 4 = -1.7808

**You may notice that the approximations did equal 3 exactly.  Recall that the square root of 3 is an irrational number and can not be expressed exactly as a decimal. You can get closer by using more decimal places.

Solve for x:   x2 + 5x + 3 =  0     

[Solution]

  
Solve for x:   -3x2 – 2x + 4 = 0     Quadratic Formula Method

Solve

Step

Check 

  a = -3, b = -2, c = 4

 Identify coefficients

If x  = -1.53518, then

-3(-1.53518)2 – 2(-1.53518) +4 = 0

   -7.07033 + 3.07036 + 4 = 0

0.00003 » 0
**Rounding error

 

If x  = 0.86852, then

-3(0.86852)2 – 2(0.86852) + 4 = 0

   -2.26298 – 1.73704 + 4 = 0

-0.00002 » 0
**Rounding error

Substitute values for a, b , and c

Simplify 
Simplify 
Take out the perfect square factor. 52 = 4×13  Square root of 4 is 2.

(exact)

(exact)

Approximations...
  x » (1 + 3.60555)/ (-3) = -1.53518
 x » (1 – 3.60555)/ (-3) = 0.86852

     Solve for x:  2x2 – 4x – 3 = 0  

[Solution]

  
  Cost Equation:  Use the cost equation, C = 0.5x2 + 20x + 4000, to find the number units x that a manufacturer can produce for the cost C = $12,000.  Round to nearest integer.
 We want to find the value(s) for x when   0.5x2 + 20x + 4000 = 12,000

0.5x2 + 20x + 4000 = 12,000

0.5x2 + 20x – 8000 = 0

 Set equal to 0 by subtracting 12,000 from both sides

Use the quadratic formula since factoring would be very difficult.
Substitute values for a, b , and c

Units must be positive!!!

Approximations...
x » -20 + 128 = 108

x » -20 – 128 = -148

There are 2 distinct answers.

The cost to manufacture 108 units is about  $12,000

Check 

C = 0.5(108)2 + 20(108) + 4000 = 5832 + 2160 + 4000 = 11,992

    Cost Equation:  Use the cost equation, C = 0.42x2 + 200x + 9,000, to find the number of units x that a manufacturer can produce for the cost C = $20,000.  Round to nearest integer.    

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada