The Quadratic Formula by Example -- Part 1.4 --
 ax2 + bx + c = 0 Memorize the formula.  See your book for proof. 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 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... 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  Approximations... x ≈ -20 + 405 = 385 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 = 0.42x2 + 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]  