QUADRATIC FUNCTIONS QUESTIONS AND ANSWERS

About "Quadratic functions questions and answers"

Quadratic functions questions and answers :

Here we are going to see some practice questions on quadratic functions.

Question 1 :

If the difference of the roots of the equation 2x2 − (a + 1)x + a −1 = 0 is equal to their product, then prove that a = 2.

Solution :

Let α and β be two roots of the given quadratic equation,

Given that :

α - β = α β

From the given quadratic equation, let us find sum and product of roots.

2x2 − (a + 1)x + a −1 = 0

α + β = -b/a

α + β = (a + 1)/2

α β = c/a

α β = (a - 1)/2

Formula for α - β = √(α + β)2 - 4αβ

√(α + β)2 - 4αβ  =  αβ

√((a+1)/2)2 - 4(a-1)/2  =  (a - 1)/2

√((a+1)/2)2 - 2(a-1)  =  (a-1)/2

√((a2 + 2a + 1) - 8(a-1))/2  =  (a-1)/2

√[(a2+2a+1)-8a+8]  =  a-1

√(a- 6a + 9)  =  a - 1

Taking squares on both side, we get

a- 6a + 9  =  (a - 1)2

a- 6a + 9  =  a2 - 2a + 1

-6a + 2a  =  1 - 9

-4a  =  -8

a  =  2

Hence proved.

Question 2 :

Find the condition that one of the roots of ax2 + bx + c may be (i) negative of the other, (ii) thrice the other, (iii) reciprocal of the other.

Solution :

 ax2 + bx + c  

Let α and β be the roots of a quadratic equation

(i)  α  =  -β (one root is negative of other), β  =  β

Sum of roots  =  α  + β  =  -b/a

-β + β  =  -b/a

0  =  -b/a

b = 0 is the required condition.

(ii)  α  =  3β (one root is thrice the other), β  =  β

Sum of roots (α+β) =  -b/a

 + β  =  -b/a

4β  =  -b/a

β  =  -b/4a  --(1)  

Product of roots αβ = c/a

 + β  =  c/a

2  =  c/a  ---(2)

Applying the value of β int the 2nd equation, we get

3(-b/4a)2  =  c/a

3b2/16a2  =  c/a

3b=  16a2c/a

3b2  =  16ac

(iii) reciprocal of the other

α  =  1/β (one root is reciprocal of other), β  =  β

Sum of roots (α+β) =  -b/a

1/β + β  =  -b/a

Product of roots (αβ)=c/a

(1/β) ⋅ β  =  c/a

1  =  c/a

a  =  c is the required condition.

After having gone through the stuff given above, we hope that the students would have understood "Quadratic functions questions and answers"

Apart from the stuff given above, if you want to know more about "Quadratic functions questions and answers".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Problems on Finding Derivative of a Function

    Mar 29, 24 12:11 AM

    Problems on Finding Derivative of a Function

    Read More

  2. How to Solve Age Problems with Ratio

    Mar 28, 24 02:01 AM

    How to Solve Age Problems with Ratio

    Read More

  3. AP Calculus BC Integration of Rational Functions by Partical Fractions

    Mar 26, 24 11:25 PM

    AP Calculus BC Integration of Rational Functions by Partical Fractions (Part - 1)

    Read More