1. Zeroes of a Quadratic Polynomial 2. Roots of Quadratic Equations 3. Formation of a Quadratic Equation

**Quadratic
Equations**

Arab mathematician
Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed
for his book ‘Liber Embadorum’ published in 1145 AD(CE) which is the first book
published in Europe to give the complete solution of a quadratic equation.

For a period of more
than three thousand years beginning from early civilizations to current times,
humanity knew how to solve a general quadratic equation in terms of its
co-efficients by using four arithmetical operations and extraction of roots.
This process is called “Solving by Radicals”. Huge amount of research has been
carried to this day in solving various types of equations.

An expression of degree *n*
in variable *x* is *a*_{0}x^{n}+ *a*_{1}x^{n−1}+*a*_{2}*x*^{n−2}+ ... +*a*_{n-1}*x* +*a*_{n}
where *a*_{0}* *≠0* *and* a*_{1}*
*,* a*_{2},...*a _{n }*are real numbers.

In particular an
expression of degree 2 is called a Quadratic Expression which is expressed as *p*(*x*)
= *ax* ^{2 }+*bx* +*c*, *a *≠0* *and* a, b,
c *are real numbers.

Let *p*(*x*)
be a polynomial. *x=a* is called zero of *p*(*x*) if *p*(*a*)
= 0

For example, if *p*(*x*)
= *x*^{2}–2*x–8* then *p*(–2)=4+4–8 = 0 and *p*(4) =
16– 8 –8 = 0

Therefore –2 and 4 are
zeros of the polynomial *p*(*x*) = *x*^{2}–2*x–8.*

Let *ax*^{2}
+ *bx* +*c* = 0, (*a* ≠ 0) be a quadratic equation. The values of
*x* such that the expression *ax* 2 + *bx* + *c* becomes
zero are called roots of the quadratic equation *ax *^{2}* *+* bx *+*c
*=* *0* *.

We have, *ax *^{2}*
*+* bx *+* c *=* *0

If *a* and *b*
are roots of a quadratic equation *ax*^{2} + *bx* +*c* =
0 then

Since, (*x* - *a*)
and (*x* - *b*) are factors of *ax*^{2} + *bx* +*c*
= 0 ,

We have (*x* - *α*)(*x*
- *β*) = 0

Hence, *x *^{2}*
*−* *(*α *+* β*)*x *+* αβ *= 0

That is, *x*^{2} - (sum of roots) *x* + product of roots = 0 is the general form of
the quadratic equation when the roots are given.

Find the zeroes of the
quadratic expression** ***x*^{2}** **+** **8*x*** **+** **12.

Let *p*(*x*)= *x*^{2}
+ 8*x* + 12 =(*x*+2)(*x*+6)

*p*(–2)= 4 – 16 + 12=0

*p*(–6)= 36 – 48 + 12=0

Therefore –2 and –6 are
zeros of *p*(*x*)= *x*^{2} + 8*x* + 12

Write down the quadratic
equation in general form for which sum and** **product of the roots are given below.

(i) 9, 14

(ii) – 7/2 , 5/2

(iii) – 3/5 , - 1/2

(i) General form of the
quadratic equation when the roots are given is *x*^{2}* *-*
*(sum of the roots ) *x *+ product of the roots = 0

*x *^{2} −* *9*x *+*
*14 = 0

Therefore, 10*x* ^{2}
+ 6*x* − 5 = 0.

Find the sum and product
of the roots for each of the following quadratic equations :

(i) *x* ^{2}
+ 8*x* − 65 = 0

(ii) 2*x*^{2}
+ 5*x* + 7 = 0

(iii) *kx *^{2}*
*−* k *^{2}* x *−* *2*k *^{3}* *=*
*0

Let* **a*** **and

*x *^{2}* *+* *8*x *−* *65*
*=* *0

*a *= 1,* b *= 8,*
c *= –65

α + *β* = −*b*/*a*
= –8 and *αβ* = *c/a* = –65

α +* β *= −8* *;
*αβ *= −65

(ii) 2*x* ^{2}
+ 5*x* + 7 = 0

*a *= 2,* b *= 5,*
c *= 7

(iii)*
kx* ^{2} − *k*
^{2} *x* − 2*k* ^{3} = 0

*a *=* k*,* b *=*
*-*k*^{2}* *,* c *= –2*k*^{3}

Tags : Expression, Zeroes, Roots, Formation, Example, Solution | Algebra , 10th Mathematics : UNIT 3 : Algebra

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10th Mathematics : UNIT 3 : Algebra : Quadratic Equations | Expression, Zeroes, Roots, Formation, Example, Solution | Algebra

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