CH – 1
QUADRATIC EQUATION
CLASS 10 TH
QUADRATIC
EQUATIONS
The polynomial of degree
two is called quadratic polynomial and equation corresponding to a quadratic
polynomial P(x) is called a quadratic equation in variable x.
Thus, P(x) = ax2
+ bx + c =0, a ≠ 0, a, b, c ∈ R is known as the standard form of
quadratic equation.
There are two types of quadratic equation.
(i) Complete quadratic equation : The equation ax2 + bx + c 0 where a ≠ 0, b ≠ 0,c ≠ 0
(ii) Pure quadratic equation : An equation in the form of ax2 = 0, a ≠ 0, b = 0, c = 0
(i) Complete quadratic equation : The equation ax2 + bx + c 0 where a ≠ 0, b ≠ 0,c ≠ 0
(ii) Pure quadratic equation : An equation in the form of ax2 = 0, a ≠ 0, b = 0, c = 0
ZERO OF A
QUADRATIC POLYNOMIAL
The value of x for which
the polynomial becomes zero is called zero of a polynomial
For instance,
1 is zero of the
polynomial x2 — 2x + 1 because it become zero at x = 1.
SOLUTION
OF A QUADRATIC EQUATION BY
FACTORISATION
A real number x is called
a root of the quadratic equation ax2 + bx + c =0, a 0 if aα2
+ bα + c =0.In this case, we say x = α is a solution of the quadratic
equation.
NOTE:
1. The zeroes of the
quadratic polynomial ax2 + bx + c and the roots of the quadratic
equation ax2 + bx + c = 0 are the same.
2. Roots of quadratic equation ax2 + bx + c =0 can be found by factorizing it into two linear factors and equating each factor to zero.
2. Roots of quadratic equation ax2 + bx + c =0 can be found by factorizing it into two linear factors and equating each factor to zero.
SOLUTION
OF A QUADRATIC EQUATION BY COMPLETING THE SQUARE
By adding and subtracting a suitable constant, we club the x2 and x
terms in the quadratic equation so that they become complete square, and solve
for x.
In fact, we can convert
any quadratic equation to the form (x + a)2 — b2 = 0 and
then we can easily find its roots.
DISCRIMINANT
The expression b2
— 4ac is called the discriminant of the quadratic equation.
SOLUTION
OF A QUADRATIC EQUATION BY DISCRIMINANT METHOD
Let quadratic equation is
ax2 + bx + c = 0
Step 1.
Find D = b2 — 4ac.
Find D = b2 — 4ac.
Step 2.
(i) If D > 0, roots are
given by
x = -b + √D / 2a , -b – √D
/ 2a
(ii) If D = 0 equation has
equal roots and root is given by x = -b / 2a.
(iii) If D < 0,
equation has no real roots.
ROOTS OF
THE QUADRATIC EQUATION
Let the quadratic equation
be ax2 + bx + c = 0 (a ≠ 0).
Thus, if b2 —
4ac ≥ 0, then the roots of the quadratic
—b ± √b2 — 4ac
/ 2a equation are given by
QUADRATIC
FORMULA
—b ± √b2 — 4ac / 2a is known as the quadratic formula
—b ± √b2 — 4ac / 2a is known as the quadratic formula
which is useful for
finding the roots of a quadratic equation.
NATURE OF
ROOTS
(i) If b2 — 4ac
> 0, then the roots are real and distinct.
(ii) If b2 —
4ac = 0, the roots are real and equal or coincident.
(iii) If b2 — 4ac <0, the roots are not real (imaginary
roots)
FORMATION OF QUADRATIC EQUATION WHEN TWO ROOTS ARE GIVEN
If α and β are
two roots of equation then the required quadratic equation can be formed as x2
— (α + β)x + αβ =0
NOTE :
Let α and β be two roots
of the quadratic equation (ax2 + bx + c = 0 then
Sum of
Roots: – the coefficient of x / the coefficient t of x2
⇒ α + β = – b / a
Product
of Roots :
αβ = constant term / the
coefficient t of x2 ⇒ αβ = c / a]
METHOD OF
SOLVING WORD PROBLEMS
Step 1: Translating
the word problem into Mathematics form (symbolic form) according to the given
condition
Step 2 : Form the
word problem into Quadratic equations and solve them.
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