QUADRATIC EQUATION - NATURE OF ROOTS

QUADRATIC EQUATION - NATURE OF ROOTS

For a quadratic equation ax² + bx + c = 0 , D = b² - 4ac is called the discriminant.

• IF D > 0 & then the roots are real and distinct.

• IF D = 0 & then the roots are real and equal.

• IF D < 0 & then the roots are not real.

NATURE OF ROOTS

• IF D > 0 & D is a perfect square then two roots are rational

NATURE OF ROOTS

• IF D > 0 & D is not a perfect square then two roots are irrational

Multiple Choice

If the roots of x² - 5x + a = 0 are equal, then a =

• $\frac{25}{4}$

• $\pm\frac{25}{4}$

• $-\frac{25}{4}$

• none of the above

Multiple Choice

If the roots of px² + qx + 1 = 0 are equal, then

• q² - 4p = 0

• q² + 4p = 0

• p² - 4p = 0

• p² + 4q = 0

Multiple Choice

If the roots 3x² + kx + 12 = 0 are equal, then k =

• ± 12

• 12

• -12

• 0

Multiple Choice

The roots of the quadratic equation x² - 6x + 10 = 0 are

• equal

• not real

• irrational

• real

Multiple Choice

The quadratic equation 4x² + 8x - (2p - 1) = 0 has real roots.

What is the range of values of p?

• $p>-\frac{3}{2}$

• $p<-\frac{3}{2}$

• $p\ge-\frac{3}{2}$

• $p\le-\frac{3}{2}$

Multiple Choice

 $If\ x^2+5px+16=0\ has\ no\ real\ roots,\ then$  

•  $p>\frac{8}{5}$  

•  $-\frac{8}{5}<p<\frac{8}{5}$  

•  $p<-\frac{8}{5}$  

• none of the above

Multiple Choice

For ax²+ bx + c = 0, which of the following statement is wrong?

• If b² – 4ac is a perfect square, the roots are rational.

• If b² = 4ac , the roots are real and equal.

• If b² – 4ac is negative, no real roots exist.

• If b² = 4ac , the roots are real and unequal.

Multiple Choice

The roots of the equation 9x² – bx + 81 = 0 will be equal, if the value of b is

• ± 9

• ± 18

• ± 27

• ± 54

Multiple Choice

Which of the following has real roots?

• $x^2\ +\ 2\sqrt{3}x\ -\ 1\ =0$

• $x^2\ +\ 2x\ +\ 4\ =\ 0$

• $x^2\ -\ x\ +\ 1\ =\ 0$

• $3x^2\ -\ 2x\ +\ 2\ =\ 0$

Multiple Choice

If px² + qx + r = 0 has equal roots, value of r will be

• $\frac{q^2}{4p}$

• $-\frac{q^2}{4p}$

• $\frac{4p}{q^2}$

• none of these

Multiple Choice

Positive value of p for which equation x² + px + 64 = 0 and x² – 8x + p = 0 will both have equal roots will be

• p ≥ 16

• p ≤ 16

• p = 16

• none of these

Multiple Choice

Write the quadratic equations whose roots have the given sum and product, sum = - 8, product = 5.

• $x^2\ +8x\ +\ 5\ =\ 0$

• $x^2\ -\ 8x\ -\ 5\ =\ 0$

• $x^2\ -\ 8x\ +\ 5\ =\ 0$

• $x^2\ +\ 5x\ -\ 8\ =\ 0$

Multiple Choice

Find the value of k so that the equation $kx^2\ -\ 24x\ +\ 16\ =\ 0$  has one real root.

• -9

• 9

• 18

• -18

Multiple Select

Determine the sum and product of the roots given the quadratic equation 13x² -26x +39 = 0.

• S: -2 and P: -3

• S: 2 and P: -3

• S: -2 and P: 3

• S: 2 and P: 3

Multiple Select

Write the quadratic equation whose roots are

 $-2\sqrt{5}\ and\ 2\sqrt{5}$  .

•  $x^2-\sqrt{5}x-4=0$  

•  $x^2+\sqrt{5}x+4=0$  

•  $x^2-4=0$  

•  $x^2-20=0$  

Multiple Select

Write the quadratic equation whose roots are

 $\sqrt{7}and\ -\sqrt{7}$  .

•  $x^2+49=0$  

•  $x^2-49=0$  

•  $x^2-7=0$  

•  $x^2+7=0$  

Multiple Choice

What is the sum and product of the roots of x² + 4x + 3 = 0?

• sum = 4; product = 3

• sum = - 4; product = 3

• sum = - 4; product = - 3

• sum = 4; product = - 3

Multiple Choice

What is the PRODUCT of the roots of the quadratic equation x² - 5x + 6 = 0?

• -5

• -6

• 5

• 6

Multiple Choice

What is the SUM of the roots of the quadratic equation x² - 5x + 6 = 0?

• -5

• -6

• 5

• 6

Multiple Choice

What is the formula in finding the PRODUCT of the roots of quadratic equation?

•  $\frac{-b}{a}$  

•  $\frac{c}{a}$  

Multiple Choice

What is the formula in finding the SUM of the roots of quadratic equation?

•  $\frac{-b}{a}$  

•  $\frac{c}{a}$  

Common new roots

 $\cdot\propto+1\ ,\ \beta+1$  
 $\cdot\frac{x}{\propto},\ \frac{x}{\beta}$  where x is a constant 
 $\cdot\propto-1\ ,\ \beta-1$  
 $\cdot\frac{1}{\propto},\ \frac{1}{\beta}$  
 $\cdot\frac{\propto}{\beta},\ \frac{\beta}{\propto}$  

New Roots

Steps:

1. State coefficients a,b,c .
2. Find  $\propto+\beta,\ \propto\beta$  
3. For new roots find: sum of roots and product of roots.
4. Substitute  $\propto+\beta,\ \propto\beta\$  from step 2 into  $x^2-\left(sum\right)x+\ product\ =0$  
N.B your answer will be a quadratic equation.

Examples of the different operations.

If  $\propto and\ \beta$  are the roots of the equation  $x^2-3x-18=0$  find:

 a.  $\propto+\beta\ \ \ \ b.\propto\beta\ \ \ \ c.\propto^2\beta^2\ \ \ d.\propto^2\beta+\beta^2\propto\$ 

 a) $\propto+\beta=\frac{-b}{a}=\frac{-\left(-3\right)}{1}=3$  

b) $\propto\beta=\frac{c}{a}=\frac{-18}{1}=-18$  

c)  $\propto^2\beta^2\Longrightarrow\left(\propto\beta\right)^2=\left(-18\right)^2=324$  
d)  $\propto^2\beta+\beta^2\propto\Longrightarrow\propto\beta\left(\propto+\beta\right)=-18\left(3\right)=-54$  

Operations with alpha and beta.

1.  $\propto+\beta=\frac{-b}{a}$    
2.  $\propto\beta=\frac{c}{a}$  
3.  $\propto^2\beta^2=\left(\propto\beta\right)^2$  
4.  $\propto^2\beta+\beta^2\propto=\propto\beta\left(\propto+\beta\right)$  
5. $\frac{x}{\propto}+\frac{x}{\beta}=\frac{x\propto+x\beta}{\propto\beta}=\frac{x\left(\propto+\beta\right)}{\propto\beta}$  where x is a constant (e.g 2,3,4,5)

6.  $\left(\propto+\beta\right)^2=squared\ of\ \propto+\beta$  
7.  $\frac{\propto}{\beta}+\frac{\beta}{\propto}=\frac{\propto^2+\beta^2}{\propto\beta}=\ \frac{\left(\propto+\beta\right)^2-2\propto\beta}{\propto\beta}$  

Example 2.

If  $\propto and\$  are the roots of the equation  $2x^2-5x+3=0$  find:

 1.  $\propto+\beta$   2.  $\propto\beta$  

 $2x^2-5x+3=0$  
 $a=2\ \ b=-5\ \ c=3$  

 $\propto+\beta=\frac{-b}{a}=\frac{-\left(-5\right)}{2}=\frac{5}{2}$  
 $\propto\beta=\frac{c}{a}=\frac{3}{2}$  

example 1.

If  $x^2-2x-8=0\ has\ roots\propto and\ \beta$  find: a. $\propto+\beta\ \ b.\propto\beta$  

 $x^2-3x-8=0$  
 $a=1\ \ b=-2\ \ c=-8$  

a)  $\propto+\beta=\frac{-b}{a}=\frac{-\left(-2\right)}{1}=\frac{2}{1}=2$  
b)  $\propto\beta=\frac{c}{a}=\frac{-8}{1}=-8$  

Representation of the relationships.

Find the values of x if  $x^2+4x-12=0$  

 $x^2+4x-12=0$  (factorize)
 $\left(x+6\right)\left(x-2\right)=0$  
 $\therefore x=-6\ or\ x=2$  

Let roots be  $\propto\ \&\ \beta$  
 $\propto=-6\ \ \beta=2$           Sum of Roots :   $\propto+\beta=-6+2=-4$                                                   Product of Roots :  $\propto\beta=-6\left(2\right)=-12$  
Therefore the relationship between the roots and coefficients is   $\propto+\beta=\frac{-b}{a}\ \ \&\ \ \propto\beta=\frac{c}{a}$  

Roots And Coefficients

 $ax^2+bx+c=0$  
The coefficients are a, b and c.

 $\propto\ \beta$  - these are called roots
 $\propto-\ alpha\ \ \ \beta-beta$  
Note: the symbol for alpha can also be represented by  $\alpha$  

There are relationships between coefficients of a quadratic equation and the roots alpha and beta. The relationship is based on the Sum of the Roots and the Product of the roots.