Angstrom Care 培苗社 Quadratic Equation II 積極推崇『我要學』的心態， 糾正『要我學』的被動心態。 www.AngstromCare.com

Quadratic Equations, Quadratic Functions and Absolute Values www.AngstromCare.com

Solving a Quadratic Equation by factorization by graphical method by taking square roots by quadratic equation by using completing square www.AngstromCare.com

By factorization roots (solutions) www.AngstromCare.com

By graphical method y roots x O www.AngstromCare.com

? By taking square roots A quadratic equation must contain two roots. www.AngstromCare.com

By taking square roots www.AngstromCare.com

Solving a Quadratic Equation by the quadratic Formula www.AngstromCare.com

By quadratic equation www.AngstromCare.com

a = 1 b = -7 c = 10 www.AngstromCare.com

In general, a quadratic equation may have : (1) two distinct (unequal) real roots (2) one double (repeated) real root (3) no real roots www.AngstromCare.com

Two distinct (unequal) real roots x-intercepts www.AngstromCare.com

One double (repeated) real roots x-intercept www.AngstromCare.com

No real roots no x-intercept www.AngstromCare.com

Nature of Roots www.AngstromCare.com

△ = b2 - 4ac Since the expression b2 - 4ac can be used to determine the nature of the roots of a quadratic equation in the form ax2 – bx + c = 0, it is called the discriminant of the quadratic equation. www.AngstromCare.com

Two distinct (unequal) real roots △ = b2 - 4ac > 0 x-intercepts www.AngstromCare.com

One double (repeated) real roots △ = b2 - 4ac = 0 x-intercept www.AngstromCare.com

No real roots △ = b2 - 4ac < 0 no x-intercept www.AngstromCare.com

Solving a Quadratic Equation by Completing the Square www.AngstromCare.com

Solving a Quadratic Equation by Completing the Square www.AngstromCare.com

Relations between the Roots and the Coefficients www.AngstromCare.com

If α and β(p and q, x1 and x2) are the roots of ax2 + bx + c = 0, then sum of roots = α + β and product of roots = αβ www.AngstromCare.com

Forming Quadratic Equations with Given Roots www.AngstromCare.com

x = 2 or x = -3 x – 2 = 0 or x + 3 = 0 (x – 2)(x + 3) = 0 Forming Quadratic Equations with Given Roots In S.3, when α = 2 and β = -3 x = 2 or x = -3 x – 2 = 0 or x + 3 = 0 (x – 2)(x + 3) = 0 x2 + x – 6 = 0 x2 – (sum of the roots)x + (product of roots) = 0 www.AngstromCare.com

Linear Functions and Their Graphs www.AngstromCare.com

www.AngstromCare.com

y c＞0 x O www.AngstromCare.com

y x O c＜0 www.AngstromCare.com

Linear Functions www.AngstromCare.com

www.AngstromCare.com

y m＞0 c＞0 c x O www.AngstromCare.com

y m＞0 c＜0 x O c www.AngstromCare.com

y c x O m＜0 c＞0 www.AngstromCare.com

y x O c m＜0 c＜0 www.AngstromCare.com

y O x c m＜0 c＝0 www.AngstromCare.com

www.AngstromCare.com

Open upwards Open upwards (a＞0) Vertex Line of symmetry www.AngstromCare.com

Vertex Open downwards Line of symmetry (a＞0) www.AngstromCare.com

Vertex (Turning point) Local (Relative) Maximum point (max. pt.) Local (Relative) Minimum point point (mini. pt.) www.AngstromCare.com

y = ax2 www.AngstromCare.com

y y = ax2 (a＞0) x O www.AngstromCare.com

y y = ax2 + bx + c b2 - 4ac＞0 2 real roots (a＞0) (c＜0) x roots O www.AngstromCare.com

y y = ax2 + bx + c b2 - 4ac＝0 repeated roots (a＞0) (c＞0) x root O www.AngstromCare.com

y y = ax2 + bx + c B2 - 4ac＜0 No real roots (a＞0) (c＞0) x No intercept www.AngstromCare.com

www.AngstromCare.com

mini value of the function = -1 mini point = (-2, -1) Finding the turning point of a Quadratic Function by Completing the Square Because a = +ve, there exists a minimum point. mini value of the function = -1 mini point = (-2, -1) www.AngstromCare.com

Absolute Values www.AngstromCare.com

Let x be any real number. The absolute value of x, denoted by | x |, is defined as x if x ≧ 0. -x if x < 0. eg. | 5 | = 5, | 0 | = 0, | -5 | = 5 www.AngstromCare.com

For all real numbers x and y, www.AngstromCare.com

If | x | = a, where a ≧0, then x = a or x = -a Generalization If | x | = a, where a ≧0, then x = a or x = -a www.AngstromCare.com

Thank you www.AngstromCare.com