Mathematics for
Computer Graphics
Linear & Nonlinear Equation
Linear Equation
• A linear equation is an
algebraic equation in
which each term is either
a constant or the product
of a constant and (the
first power of) a single
variable.
Nonlinear Equation
• A nonlinear system of equations is a set of
simultaneous equations in which the
unknowns appear as variables of a
polynomial of degree higher than one or in
the argument of a function which is not a
polynomial of degree one
Quadratic Equation
• Quadratic equation is an equation of degree 2
with 3 coefficient a, b and c
ax2 + bx + c = 0
Root of Quadratic Equation
Given the quadratic equation where a ≠ 0;
ax2 + bx + c = 0
• Step 1 : subtract c from both sides:
ax2 + bx = -c
• Step 2: divide both sides by a:
x2 + bx = -c
a
a
• Step 3: add b2/4a2 to both sides:
x2 + bx + b2 = -c + b2
a 4a2 a 4a2
• Step 4 : factorize the left side:
x + b 2 = -c + b2
2a
a 4a2
Root of Quadratic Equation
• Step 5: make 4a2 the common denominator for the right side:
x + b 2 = -4ac + b2
2a
4a2
• Step 6: take the square root of both sides:
x + b = ±√ b2 – 4ac
2a
2a
• Step 7: subtract b/2a from both sides:
x = ±√ b2 – 4ac - b
2a
2a
• Step 8: rearrange the right side:
x = -b ±√ b2 – 4ac
2a
Relation between roots &
coefficients of a quadratic equation
• Roots of quadratic equation, x1 and x2
x1 = -b +√ b2 – 4ac and x2 = -b -√ b2 – 4ac
2a
2a
• Sum of the roots
x1 + x2 = -b +√ b2 – 4ac + -b -√ b2 – 4ac
2a
2a
= -2b/2a
= - b/a
• Product of the roots
x1x2= -b +√ b2 – 4ac x -b -√ b2 – 4ac
2a
2a
= b2 – (b2 – 4ac )
4a2
= c/a
Quadratic Properties
• Law of Trichotomy
a < 0 and the parabola opens down.
a = 0 and the function is not quadratic.
a > 0 and the parabola opens up.
b2 - 4ac < 0 and the parabola has no xintercepts.
b2 - 4ac = 0 and the parabola has exactly
one x-intercept.
b2 - 4ac > 0 and the parabola has two xintercepts.
Example
• Given a quadratic equation, 5x2 - 3x - 10 = 0.
Find the roots, the sum and the product of the
roots.
• Solution
x1 = -b +√ b2 – 4ac = ? ,
2a
x2 = ?
x1 + x2 = -b/a = -(-3)/5 = 3/5
x1.x2 = c/a = -10/5 = - 2
Nonlinear Equation:
Cubic Equation