The Quadratic Formula by Example 
 Part 1.4  
ax^{2} + bx + c = 0 
Memorize the formula. See your book for proof.
Solve for x: 2x^{2} + 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 If x = 1.7808, then 2(1.7808)^{2} + 3(1.7808)  1 = 0 6.3425 
5.3424  1 = 0 

Substitute values for a, b , and c  

Simplify  

There are 2 distinct answers.  
(exact) 
(exact)  
Approximations... 

**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: x^{2} + 5x + 3 = 0  [Solution]  


Solve for x: 3x^{2} – 2x + 4 = 0 Quadratic Formula Method  
Solve 
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
1
If x = 0.86852, then 3(0.86852)^{2} – 2(0.86852) + 4 = 0 2.26698 – 1.73704 + 4 = 0 0.00002
1 
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)  
Approximations... 

Solve for x: 2x^{2} – 4x – 3 =0  [Solution] 



Cost Equation: Use the cost equation, C = 0.5x^{2} + 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 0.5x^{2} + 20x + 4000 = 12,000  
0.5x^{2} + 20x + 4000 = 12,000 0.5x^{2} + 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 



Approximations... x ≈ 20 – 405 = 425 


There are 2 distinct answers.  The cost to manufacture 385 units is $12,000 
Cost Equation: Use the cost equation, C = 0.42x^{2} + 200x + 9,000, to find the number units x that a manufacturer can produce for the cost C = $20,000. Round to nearest integer.  
[Solution] 

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joe mcdonald
Last Update 09.23.2010