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 If x  = -1.7808, then 2(-1.7808)2 + 3(-1.7808) - 1 = 0    6.3425 - 5.3424 - 1 = 0 0.0001 » 1 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  If x  = 0.86852, then-3(0.86852)2 – 2(0.86852) + 4 = 0    -2.26298 – 1.73704 + 4 = 0 -0.00002 » 0 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