**Algebraic Expressions Questions and Answers – Formulas & Tricks**

- (a+b)
^{2}= a^{2}+b^{2}+2ab - (a-b)
^{2}= a^{2}+b^{2}-2ab - a
^{2}-b^{2}= (a+b)(a-b) - (a+b)
^{3}= a^{3}+b^{3}+3ab(a+b) - (a-b)
^{3}= a^{3}-b^{3}-3ab(a-b) - a
^{3}+b^{3}= (a+b)(a^{2}+b^{2}-ab) - a
^{3}-b^{3}= (a-b)(a^{2}+b^{2}+ab) - (a+b+c)
^{2}=a^{2}+b^{2}+c^{2}+2(ab+bc+ca) - (a+b+c)
^{3}=a^{3}+b^{3}+c^{3}+3(a+b)(b+c)(c+a) - a
^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca) = (1/2)(a+b+c)[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}]

**Tricks to Remember**

**Quadratic Equations:**

- An equation of the type ax
^{2}+bx+x = 0 is called the Quadratic Equation - The highest power of the variable is called the degree of the equation
- An equation will have as many solutions as its degre. If an equation is of n degree, it will have ‘n’ solutions.
- The solution of an equation is the value by which the equation is satisfied. The value of the solution of an equation is also called the roots of the equation. The quadratic equation has two solutions

**Solving Quadratic Equation:**

Any Quadratic Equation can be solved by **Factor Method or Formula**

**Factor Method**: 1^{st}find the factors of the given equation making the right-hand side equal to zero and then by equating the factors to zero, We get the values of the variable.**Formula:**Consider a quadratic equation ax2+bx+c=0, finding the roots of the equation, we use the following formula:

**Algebraic Expressions Questions and Answers With Detailed Explanation – Quantitative Aptitude**

## Algebraic Expressions

Question 1 |

1/8 | |

-(1/8) | |

8/3 | |

-(8/3) |

Question 1 Explanation:

**Answer: Option B**

__Explanation:__

From the Given Equation

(2x + 3) (3x + 1) = (3x - 1)(2x - 1)

=> 6x

^{2}+ 11x + 3 = 6x

^{2}-5x+1

=16x + 2

=16x = -2

x = -(2/16)

**x = -(1/8)**

Question 2 |

Question 2 Explanation:

**Answer: Option A**

__Explanation:__

Question 3 |

The equation whose roots are 4 and 5, is

x ^{2}+9x-20 = 0 | |

x ^{2}+9x+20 = 0 | |

x ^{2}-9x+20 = 0 | |

x ^{2}-9x+20 = 0 |

Question 3 Explanation:

**Answer: Option C**

__Explanation:__

The required equation is:

x

^{2}- (Sum of Roots)x + product of roots = 0

=> X

^{2}-(4+5)x+4*5 = 0

**X**

^{2}-9x+20 = 0Question 4 |

The roots of the equation (x + 3) (x - 3) = 160 are

± 13 | |

13, 13 | |

± 12 | |

12, 12 |

Question 4 Explanation:

**Answer: Option A**

__Explanation:__

(x + 3) (x - 3) = 160

x

^{2}- 9 = 160

x

^{}2 = 169

x =+13, -13

Question 5 |

If the equation 4x

^{2}+x(p+1)+1 = 0 has exactly two equal roots, one of the value of p is5 | |

-3 | |

0 | |

3 |

Question 5 Explanation:

**Answer: Option D**

__Explanation:__

Roots are equal if discriminant

b

^{2}- 4ac = 0

In given equation:

(P + 1)

^{2}- 4 * 4 * 1 = 0

(P + 1)

^{2 }= 16 = (4)

^{2}

P + 1 = +4, -4

P = 3 or -5

Question 6 |

The solution for

-5 | |

-1 | |

-3 | |

3 |

Question 6 Explanation:

**Answer: Option A**

__Explanation:__

The given equation:

x

^{2}-3x-4x-12 = -6x+18

x

^{2}-x-30 = 0

x

^{2}+5x-6x-30 = 0

(x+5)(x-6) = 0

**x = -5 or 6**

Question 7 |

If X

^{2}+ aX + b = 0 x^{2}+ bx + a = 0 is double the other; a, b, c are related as2a ^{2} = 9bc | |

9a ^{2} = 2bc | |

2b ^{2} = 9ac | |

9b ^{2} = 2ac |

Question 7 Explanation:

If α and β be the roots of the equation:

Then, α + β =-(b/a), and α β = (c/a)

Also, α = 2β. Then

3β= -(b/a)

β = -(b/3a)

And,

2β

Therefore 2*-(b/3a)

=> (2b

=>2b

Then, α + β =-(b/a), and α β = (c/a)

Also, α = 2β. Then

3β= -(b/a)

β = -(b/3a)

And,

2β

^{2}=(c/a)Therefore 2*-(b/3a)

^{2}= (c/a)=> (2b

^{2}/9a^{2}) = c/a=>2b

^{2}= 9acQuestion 8 |

Two Mixers and one TV cost Rs. 7000 while two T.V's and 'one mixer cost Rs. 8, 800. What is the price of one mixer (in Rupees)?

1400 | |

1300 | |

1250 | |

1100 |

Question 8 Explanation:

**Answer: Option A**

__Explanation:__

Let the price of mixer is Rs. x

And, the price of a T.V = Rs. y

As per the Question

2x + y = 7,000 .... (i)

x + 2y = 9,800 ....(ii)

x= 9800 - 2y

2(9800 - 2y) + y = 7000

19600 - 4y + y = 7000

19600 - 3y = 7000

19600 - 7000 =3y

12600 = 3y

12600 = 3y

y = 12600/3 = 4200

Putting the value of y in equation (ii)

x + 2 * 4200 = 9800

x = 9800 - 8400

**x = 1400**

Question 9 |

Today Madan is 20 years younger than his father. Ten Years ago he was one-half as old as his father. How old his father is ten years hence?

40 Years | |

50 Years | |

60 Years | |

Data inadequate |

Question 9 Explanation:

**Answer: Option C**

__Explanation:__

Let the age of Madan and his father be x and y years respectively. Then

y-x = 20 .....(i)

Ten year ago:

x=10 = (1/2) (y-10)

2(x-10) = y - 10

2x - 20 = y - 10

2x - y = -10 + 20

2x - y = 10 ....(ii)

Now, solving eq.1 and eq.2

y-x=20

-y + 2x =10

x = 30(age of Madan)

So, age of his father: y = x + 20

=30 + 20 = 50

Madan's father age after 10 Years:

y + 10 = 50 + 10 = 60 Years

Question 10 |

Presently the age of Meera's mother is four times that of Meera's. After 5 Years her mother's age will be three times that of Meera's age. What is Meera's Present age?

15 | |

10 | |

20 | |

5 |

Question 10 Explanation:

**Answer: Option B**

__Explanation:__

Let Meera's present age is x years, and that of the mother is y Years.

y = 4x

4x - y = 0 .... (i)

After five years:

y + 5 = 3 (x + 5)

3x - y = -10 ....(ii)

On solving the above two equations:

x = 10, y = 40

So, Meera's Present age = 10 Years

There are 10 questions to complete.