Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §8.2 Quadratic Equation Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

8.1 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55 Review § Any QUESTIONS About §8.1 → Complete the Square Any QUESTIONS About HomeWork §8.1 → HW-31

This is one of the MOST FAMOUS Formulas in all of Mathematics The Quadratic Formula The solutions of ax2 + bx + c = 0 are given by This is one of the MOST FAMOUS Formulas in all of Mathematics

§8.2 Quadratic Formula The Quadratic Formula Problem Solving with the Quadratic Formula

Derive Quadratic Formula - 1 Consider the General Quadratic Equation Next, Divide by “a” to give the second degree term the coefficient of 1 Where a, b, c are CONSTANTS Solve This Eqn for x by Completing the Square First; isolate the Terms involving x Now add to both Sides of the eqn a “quadratic supplement” of (b/2a)2

Derive Quadratic Formula - 2 Now the Left-Hand-Side (LHS) is a PERFECT Square Use the Perfect Sq Expression Finally Use the Sq-Root Property to solve for x Solve for x; but first let

Derive Quadratic Formula - 3 Alternatively, Start with the PERFECT SQUARE Expression Combine Terms inside the Radical over a Common Denom Take the Square Root of Both Sides

Derive Quadratic Formula - 4 Note that Denom is, itself, a PERFECT SQ Now Combine over Common Denom But this the Renowned QUADRATIC FORMULA Note That it was DERIVED by COMPLETING the SQUARE Next, Isolate x The quadratic formula was derived BEFORE Phythagorus

Example a) 2x2 + 9x − 5 = 0 Solve using the Quadratic Formula: 2x2 + 9x − 5 = 0 Soln a) Identify a, b, and c and substitute into the quadratic formula: 2x2 + 9x − 5 = 0 a b c Now Know a, b, and c

Solution a) 2x2 + 9x − 5 = 0 Using a = 2, b = 9, c = −5 Recall the Quadratic Formula → Sub for a, b, and c Be sure to write the fraction bar ALL the way across.

Solution a) 2x2 + 9x − 5 = 0 From Last Slide: So: The Solns:

Example b) x2 = −12x + 4 Soln b) write x2 = −12x + 4 in standard form, identify a, b, & c, and solve using the quadratic formula: 1x2 + 12x – 4 = 0 a b c

Example c) 5x2 − x + 3 = 0 Soln c) Recognize a = 5, b = −1, c = 3 → Sub into Quadratic Formula The COMPLEX No. Soln Since the radicand, –59, is negative, there are NO real-number solutions.

Quadratic Equation Graph The graph of a quadratic eqn describes a “parabola” which has one of a: Bowl shape Dome shape x intercepts vertex The graph, depending on the “Vertex” Location, may have different numbers of of x-intercepts: 2 (shown), 1, or NONE

The Discriminant It is sometimes enough to know what type of number (Real or Complex) a solution will be, without actually solving the equation. From the quadratic formula, b2 – 4ac, is known as the discriminant. The discriminant determines what type of number the solutions of a quadratic equation are. The cases are summarized on the next sld

Soln Type by Discriminant Discriminant b2 – 4ac Nature of Solutions x-Intercepts Only one solution; it is a real number Only one Positive Two different real-number solutions Two different Negative Two different NONreal complex-number solutions (complex conjugates) None

Example  Discriminant Determine the nature of the solutions of: 5x2 − 10x + 5 = 0 SOLUTION Recognize a = 5, b = −10, c = 5 Calculate the Discriminant b2 − 4ac = (−10)2 − 4(5)(5) = 100 − 100 = 0 There is exactly one, real solution. This indicates that 5x2 − 10x + 5 = 0 can be solved by factoring  5(x − 1)2 = 0

Example  Discriminant Determine the nature of the solutions of: 5x2 − 10x + 5 = 0 SOLUTION Examine Graph Notice that the Graph crosses the x-axis (where y = 0) at exactly ONE point as predicted by the discriminant

Example  Discriminant Determine the nature of the solutions of: 4x2 − x + 1 = 0 SOLUTION Recognize a = 4, b = −1, c = 1 Calculate the Discriminant b2 – 4ac = (−1)2 − 4(4)(1) =1 − 16 = −15 Since the discriminant is negative, there are two NONreal complex-number solutions

Example  Discriminant Determine the nature of the solutions of: 4x2 − 1x + 1 = 0 SOLUTION Examine Graph Notice that the Graph does NOT cross the x-axis (where y = 0) indicating that there are NO real values for x that satisfy this Quadratic Eqn

Example  Discriminant Determine the nature of the solutions of: 2x2 + 5x = −1 SOLUTION: First write the eqn in Std form of ax2 + bx + c = 0 → 2x2 + 5x + 1 = 0 Recognize a = 2, b = 5, c = 1 Calculate the Discriminant b2 – 4ac = (5)2 – 4(2)(1) = 25 – 8 = 17 There are two, real solutions

Example  Discriminant Determine the nature of the solutions of: 0.3x2 − 0.4x + 0.8 = 0 SOLUTION Recognize a = 0.3, b = −0.4, c = 0.8 Calculate the Discriminant b2 − 4ac = (−0.4)2 − 4(0.3)(0.8) =0.16–0.96 = −0.8 Since the discriminant is negative, there are two NONreal complex-number solutions

Writing Equations from Solns The principle of zero products informs that the this factored equation (x − 1)(x + 4) = 0 has solutions 1 and −4. If we know the solutions of an equation, we can write an equation, using the principle in reverse.

Example  Write Eqn from solns Find an eqn for which 5 & −4/3 are solns SOLUTION x = 5 or x = –4/3 x – 5 = 0 or x + 4/3 = 0 Get 0’s on one side Using the principle of zero products (x – 5)(x + 4/3) = 0 x2 – 5x + 4/3x – 20/3 = 0 Multiplying 3x2 – 11x – 20 = 0 Combining like terms and clearing fractions

Example  Write Eqn from solns Find an eqn for which 3i & −3i are solns SOLUTION x = 3i or x = –3i x – 3i = 0 or x + 3i = 0 Get 0’s on one side Using the principle of zero products (x – 3i)(x + 3i) = 0 x2 – 3ix + 3ix – 9i2 = 0 Multiplying x2 + 9 = 0 Combining like terms

WhiteBoard Work Problems From §8.2 Exercise Set 18, 30, 44, 58 Solving Quadratic Equations 1. Check to see if it is in the form ax2 = p or (x + c)2 = d. If it is, use the square root property 2. If it is not in the form of (1), write it in standard form: ax2 + bx + c = 0 with a and b nonzero. 3. Then try factoring. 4. If it is not possible to factor or if factoring seems difficult, use the quadratic formula. The solns of a quadratic eqn cannot always be found by factoring. They can always be found using the quadratic formula.

All Done for Today The Quadratic Formula

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –

Graph y = |x| Make T-table

Quadratic Equation Graph The graph of a quadratic eqn describes a “parabola” which has one of a: Bowl shape Dome shape The graph, depending on the “Vertex” Location may have different numbers of x-intercepts: 2 (shown), 1, or NONE