Solving Quadratic Equations By The Formula In Graphic Form Worksheet With Answers By Nghi H Nguyen printable pdf download

Solving Quadratic Equations By The Formula In Graphic Form Worksheet With Answers By Nghi H Nguyen

SOLVING QUADRATIC EQUATIONS BY

THE FORMULA IN GRAPHIC FORM

(By Nghi H Nguyen)

There are many methods to solve a quadratic equation in standard form: ax^2 + bx + c = 0.

If the given equation can be factored, the two best methods to solve it are “the factoring ac

method” (You Tube) and the new Diagonal Sum Method. In case the given quadratic equation

can’t be factored, the quadratic formula would be the best way to solve it.

An improved quadratic formula, called: ”The quadratic formula in graphic form”, has been

presented in the book titled: ”New methods for solving quadratic equations and inequalities”

(Amazon e-book 2010).

The graph of the quadratic function f(x) = ax^2 + bx + c is a parabola that may intercept the x-

axis at 2 points, unique point, or no point at all. This means a quadratic equation f(x) = ax^2 + bx

+ c = 0 may have 2 real roots, one double real root, or no real roots at all (complex roots).

Given a quadratic function in standard form f(x) = ax^2 + bx + c and its parabola graph drawn

on a coordinate system Ox,Oy. Suppose A is the x-intercept of the parabola axis, and suppose

the parabola intercepts the x-axis at 2 points B and C.

The quadratic formula in graphic form gives the 2 real roots x1 and x2 by this formula:

x1 = -b/2a + d/2a

and

x2 = - b/2a – d/2a

(1)

In this formula:

The quantity (–b/2a) represents the x-coordinate of the parabola axis.

The 2 quantities (d/2a) and (-d/2a) represent the 2 distances AB and AC from the

parabola axis to the two x-intercepts (real roots) of the parabola.

The quantity d can be zero, a number, or imaginary.

If d = 0: there is a double root at x = -b/2a.

If d is a number (real or radical), there are two x-intercepts, meaning two real roots.

If d is imaginary, there are no real roots.

The unknown quantity d can be computed from the constants a, b, c by this relation (2):

d^2 = b^2 – 4ac

(2)

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Solving Quadratic Equations By The Formula In Graphic Form Worksheet With Answers By Nghi H Nguyen