Sum and product of the roots of a quadratic equation.Definition of a Quadratic equation.

Contents:-

Without even finding the actual roots of a quadratic equation using the Factorization method or The Quadratic formula, we can find the sum and product of the roots, just by figuring out coefficients a,b,c of the quadratic.

We can get back the quadratic equation knowing the sum and product of roots of a quadratic.

Following is the brief of concepts covered and features of this article. Please do attempt the “Quiz” at the end.

  • The formula for Sum and product of the roots.
  • Finding the quadratic equation from the given Sum and product of the roots.
  • Roots of a quadratic equation and relation with the “x” intercepts.
  • Worked out examples.
  • Practice problems with solutions.
  • Quiz with solutions.

1. Formulas for the Sum and product of roots of a quadratic equation:-

The sum and product of roots of a quadratic equation  ax^{2}+bx+c=0.can be found from the coefficients of  x^ {2}  ,x and the constant number which are a,b,c respectively.

The Sum of the roots of a quadratic equation: Sum of the roots of a quadratic equation is given by the  formula  -\dfrac {b}{a}

The product of roots of a quadratic equation: The product of roots of a quadratic equation is given by the  \dfrac {c}{a}

2. How to find the quadratic function, given the sum and product of the roots?:-

When a quadratic function is given, we can find the sum and product of the roots using the formulas given above.

However, when we have the information about the sum and product of roots, we can find the quadratic function back.

Let us say that the sum of the roots is given as “p” and the product of roots is given as “q”.

The equation of the quadratic is given by  x^{2}-px+q=0.

3.Definition of a Quadratic Equation:

The term “quadratic equation” includes an “equal to”symbol“=”.The quadratic equation is generally represented as  ax^{2}+bx+c=0 . .

The sum and product of the roots of a quadratic equation can be found from the coefficients of the quadratic equation itself.

In a quadratic equation, there might exist values of “x” such that the value of the expression  ax^{2}+bx+c.becomes equal to “0”.

The “x” values for which the quadratic expression becomes “0”, are generally called as“The roots of the quadratic equation” OR “The solutions of a quadratic equation “OR“The‘x’ intercepts of a quadratic equation”.

Since the order of a quadratic equation is 2, the number of roots can NEVER be more than two.

A quadratic equation can have a maximum ‘Two roots’ or ‘a double root at a point’ or ‘Nil roots’.

i).Graphs of Quadratic equations and “x” intercepts:-

  • ‘NIL’ solutions/ roots of a Quadratic equation:

If a quadratic equation is having NIL solutions/roots, it implies that the graph of the quadratic neither touches nor intersects the ‘x’ axis.

Parabola not touching "x" axis

  • Single solution/roots of the quadratic equation with double root:-

If a quadratic equation has a single solution, we can conclude that there is a double root at a point on the “x” axis.

The graph just touches the “x” axis and will not intersect the x-axis.

Parabola intersecting "x" axis at a single point

 

  • Two distinct roots of a Quadratic equation: –

If there are two distinct real solutions/ roots of quadratic equation, It implies that graph of the quadratic equation intersects the “x” axis twice.

Parabolic graph intersecting "x" axis at two points

4. Worked out the examples:-

Example 1: Find the sum and product of roots of a quadratic equation.

i) x^{2}+4x +5

Solution:

Strategy Step-by-Step:-

   Step1: Identify the coefficients a,b,c of the quadratic equation.

Comparing with the standard form of quadratic equation, we observe that a=1, b=4, and c=5.

   Step2: Find the ratio -\dfrac {b}{a} for the sum of the  roots.

Therefore the sum of the  roots would be \dfrac {b}{a} =-\dfrac {4}{1} .

   Step3 :Find the ratio \dfrac {c}{a}

Product  of  roots would be \dfrac {c}{a} =\dfrac {5}{1} .

 

Example 2: what is the quadratic equation whose roots are -3, -1 and has a leading coefficient of 2 with x to represent the variable?

Solution: Given that the leading coefficient a=2 and we need to use the variable “x” to represent the quadratic function.

Given that the roots are -3,-1. Therefore the sum of the roots would be -3-1 =-4 and product of roots would be (-3)*(-1) =3

Sum of the roots of a quadratic equation is given by the  formula  -\dfrac {b}{a}.Therefore on plugging a=2 we can have

\begin{aligned}-\dfrac {b}{2}=-4\\ \Rightarrow 2\left( \dfrac {-b}{2}\right) =2\left( -4\right) \\ \Rightarrow -b=-8\\ \Rightarrow b=8\end{aligned}

Similar way, The product of roots of a quadratic equation is given by the  \dfrac {c}{a},we have product of roots as 3 .Hence we can equate both.

\begin{aligned}\dfrac {c}{a}=3\\ \Rightarrow \dfrac {c}{2}=3\\ \Rightarrow 2\left( \dfrac {c}{2}\right) =2\left( 3\right) \\ \Rightarrow c=6\end{aligned}

Since we have the coeffieicnts a=2 ,b= 8, c=6, we can plug these values into the standard form of the quadratic ax^{2}+bx+c=0..

Therefore the quadratic equation in “x” whose roots are -3, -1 and having a leading coefficient of 2  would be  2x^{2}+8x+6=0. .

 

Example 3: Find the quadratic equation, whose sum and product of roots are 5 and 4 respectively.

Step1: Identify the sum of the roots P and the product of roots Q from the problem given.

From the problem, we can say that sum of the roots is p= 5 and product of roots q=4.

Step2: Plug in the values of P and Q into the formula x^{2}-px+q=0.

Therefore the equation of the quadratic function will be  x^{2}-5x+4=0.

 

5)  Practice Problems

Find the sum and product of roots of a quadratic equation.

i) 0.5x^{2}+x +2

ii) x^{2}+4.

iii) -x^{2}+5x

 iv)-2x^{2}-6x +8

v)-4x^{2}+8

6)  Solutions to Practice problems:-

i)  0.5x^{2}+x +2 .On comparison with the standard form the quadratic equation  f\left( x\right) =ax^{2}+bx+c ., we observe that there is  a number 0.5 before the  “x^2” term default number 1 before the  “x” term and a constant number c=2.

Therefore the sum of the  roots would be – \dfrac {b}{a} =- \dfrac {1}{0.5} . = -2.

Product  of  roots would be \dfrac {c}{a} =\dfrac {2}{0.5} .=4

 

ii) x^{2}+4 .On comparison with the standard form the quadratic equation  f\left( x\right) =ax^{2}+bx+c ., we observe that there is  a default number 1 before the  “x^2” term , No “x” term which means b=0 and a constant number c=4.

Therefore the sum of the  roots would be – \dfrac {b}{a} =- \dfrac {0}{1} . = 0.

Product  of  roots would be \dfrac {c}{a} =\dfrac {4}{1} .= 4

 

iii)  -x^{2}+5x  .On comparison with the standard form the quadratic equation  f\left( x\right) =ax^{2}+bx+c ., we observe that there is  a negativebefore the  “x^2” term which means a=-1,a number 5 before the  “x” term therefore b=5 and no constant term which means c=0.

Therefore the sum of the  roots would be – \dfrac {b}{a} = – \dfrac {5}{-1} . = 5.

Product  of  roots would be \dfrac {c}{a} =\dfrac {0}{-1} .= 0.

 

iv)  -2x^{2}-6x +8 .On comparison with the standard form the quadratic equation  f\left( x\right) =ax^{2}+bx+c ., we observe that there is  a negative 2 before the  “x^2” term which means a=-2 ,a number -6 before the  “x” term therefore b=-6 and constant is c=8.

Therefore the sum of the  roots would be – \dfrac {b}{a} = – \dfrac {-6}{-2} . = -3.

Product  of  roots would be \dfrac {c}{a} =\dfrac {8}{-2} .= -4.

 

v) -4x^{2}+8 .On comparison with the standard form the quadratic equation  f\left( x\right) =ax^{2}+bx+c ., we observe that there is  a number -4 before the  “x^2” term , No “x” term which means b=0 and a constant number c=8.

Therefore the sum of the  roots would be – \dfrac {b}{a} =- \dfrac {0}{1} . = 0.

Product  of  roots would be \dfrac {c}{a} =\dfrac {8}{-4} .= -2.

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Quiz Time

QUIZ 1of3 ( Number of roots of a quadratic)

Let us have a quick quiz on concepts learned :-

QUIZ 2 of 3 (Sum and product of roots)

Let us have a quick quiz on concepts learned :-

QUIZ ( 3 of 3)(Quadratic function from sum and product of roots)

Let us have a quick quiz on concepts learned :-

 

 

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