Mathematics Reference Manual
Quadratic Equations
Forming Quadratic Polynomials
A quadratic polynomial is of the form (ax^2 + bx + c).
Example:
Form a quadratic polynomial given the roots (alpha = 3) and (beta = -2).
Sum of roots: (alpha + beta = 1)
Product of roots: (alpha beta = -6)
Quadratic polynomial: (x^2 – (alpha + beta)x + alpha beta = x^2 – x – 6)
Sum and Product of Roots
For a quadratic equation (ax^2 + bx + c = 0), the sum and product of the roots are given by:
- (alpha + beta = -frac{b}{a})
- (alpha beta = frac{c}{a})
Example:
Given the quadratic equation (x^2 – 5x + 6 = 0), find the sum and product of the roots.
Sum of roots: (alpha + beta = 5)
Product of roots: (alpha beta = 6)
Discriminant
The discriminant of a quadratic equation (ax^2 + bx + c = 0) is given by (Delta = b^2 – 4ac). The nature of roots depends on the discriminant:
- If (Delta > 0), the equation has two distinct real roots.
- If (Delta = 0), the equation has two equal real roots.
- If (Delta < 0), the equation has two complex roots.
Example:
Given the quadratic equation (x^2 – 4x + 4 = 0), determine the nature of its roots.
(Delta = 4^2 – 4 cdot 1 cdot 4 = 16 – 16 = 0)
Nature of roots: Two equal real roots
Completing the Square
Completing the square is a method used to solve quadratic equations by rewriting the quadratic expression in the form ((x – p)^2 = q).
Steps to Complete the Square
- Start with a quadratic equation in the form (ax^2 + bx + c = 0).
- Divide all terms by (a) (if (a neq 1)).
- Move the constant term (c/a) to the right side of the equation.
- Add and subtract ((b/2a)^2) to/from the left side to form a perfect square trinomial.
- Factor the perfect square trinomial.
- Solve for (x).
Example:
Solve the quadratic equation (2x^2 + 8x – 10 = 0) by completing the square.
Divide all terms by 2: (x^2 + 4x – 5 = 0)
Move the constant term: (x^2 + 4x = 5)
Add and subtract 4: (x^2 + 4x + 4 – 4 = 5 quad Rightarrow quad x^2 + 4x + 4 = 9)
Factor the left side: ((x + 2)^2 = 9)
Take the square root: (x + 2 = pm 3)
Solve for (x): (x = 1 quad text{or} quad x = -5)
Solution: The roots of the equation are (x = 1) and (x = -5).
Polynomials and Roots
Finding the Roots of a Polynomial
The roots of a polynomial are the values of (x) that satisfy the polynomial equation.
Example:
Find the roots of the polynomial (x^2 – 5x + 6).
Factorize: ((x – 2)(x – 3) = 0)
Roots: (x = 2 quad text{and} quad x = 3)
Factorization
Expressing a polynomial as a product of its factors helps in solving polynomial equations.
Example:
Factorize the polynomial (x^2 – 4).
Factorize: ((x – 2)(x + 2) = 0)
Factors: (x – 2 quad text{and} quad x + 2)
Rational Expressions
Simplifying Rational Expressions
Combining and simplifying expressions involving fractions.
Example:
Simplify the rational expression (frac{x}{x^2 – 1} + frac{1}{x – 1}).
(frac{x}{(x – 1)(x + 1)} + frac{1}{x – 1} = frac{x + 1}{(x – 1)(x + 1)} = frac{1}{x – 1})
Exponential Equations
Solving Exponential Equations
Equations involving terms with variables in the exponent.
Example:
Solve the equation (2^{x+1} = 8).
Rewrite: (2^{x+1} = 2^3)
Equate exponents: (x + 1 = 3)
Solution: (x = 2)
Radical Equations
Solving Radical Equations
Equations involving square roots or higher-order roots.
Example:
Solve the equation (sqrt{2x + 3} = x + 1).
Square both sides: (2x + 3 = (x + 1)^2)
Expand and solve: (2x + 3 = x^2 + 2x + 1 Rightarrow x^2 – 2 = 0)
Roots: (x = pm sqrt{2})
Systems of Equations
Solving Systems of Equations
Solving for multiple variables by setting up and solving multiple equations.
Example:
Solve the system of equations:
[
begin{cases}
2x + 3y = 5 \
x – y = 1
end{cases}
]
Multiply the second equation by 3 and add to the first: (2x + 3y + 3(x – y) = 5 + 3 Rightarrow 5x = 8 Rightarrow x = frac{8}{5})
Substitute (x) into the second equation: (frac{8}{5} – y = 1 Rightarrow y = frac{3}{5})
Arithmetic and Geometric Progressions
Sum of Arithmetic Progressions
Using the formula for the sum of the first (n) terms of an arithmetic sequence.
Example:
Find the sum of the first 5 terms of the arithmetic progression with first term (a = 2) and common difference (d = 3).
(S_n = frac{n}{2}(2a + (n – 1)d))
(S_5 = frac{5}{2}(2 cdot 2 + 4 cdot 3) = frac{5}{2}(4 + 12) = 40)
Logarithmic Equations
Solving Logarithmic Equations
Using properties of logarithms to solve equations involving logarithmic expressions.
Example:
Solve the equation (log_2(x + 3) = 4).
Rewrite: (x + 3 = 2^4 = 16)
Solution: (x = 13)
Symmetry and Geometry
Properties of Triangles and Circles
Using geometric properties and theorems to solve problems involving triangles and circles.
Example:
Find the hypotenuse of a right triangle with legs 3 cm and 4 cm.
Hypotenuse: (c = sqrt{3^2 + 4^2} = sqrt{9 + 16} = 5 , text{cm})
Algebraic Manipulations
Simplifying Complex Expressions
Using algebraic identities and techniques to simplify and solve complex expressions.
Example:
Simplify the expression (frac{x^2 – 1}{x – 1}).
(frac{(x – 1)(x + 1)}{x – 1} = x + 1)
Specific Concepts and Formulas Required
Roots of Unity
Using the properties of roots of unity in polynomial equations.
Example:
Find the cube roots of unity: (1, omega, omega^2), where (omega = e^{2pi i / 3}).
Sum of Inverses of Roots
For roots (alpha) and (beta), the sum of their inverses is given by (alpha^{-1} + beta^{-1} = frac{alpha + beta}{alpha beta}).
Example:
Given the quadratic equation (x^2 – 5x + 6 = 0), find the sum of the inverses of its roots.
Roots: (alpha = 2, beta = 3)
Sum of inverses: (alpha^{-1} + beta^{-1} = frac{1}{2} + frac{1}{3} = frac{3 + 2}{6} = frac{5}{6})
Example Problems
Questions
- Form a quadratic polynomial, one of whose zeroes is $2 + sqrt{5}$ and the sum of zeroes is 4.
- Form a quadratic polynomial, one of whose zeroes is $sqrt{5}$ and the product of the zeroes is $-2 sqrt{5}$.
- Determine if 3 is a zero of the polynomial (P(x) = sqrt{x^2 – 4x + 3} + sqrt{x^2 – 9} – sqrt{4x^2 – 14x + 6}).
- (alpha, beta) are zeroes of the quadratic polynomial (x^2 – (k + 6)x + 2(2k – 1)). Find the value of (k) if (alpha + beta = frac{1}{2} alpha beta).
- (m, n) are zeroes of (ax^2 – 5x + c). Find the value of (a) and (c) if (m + n = 5) and (mn = 10).
- Form a quadratic polynomial whose zeroes are (frac{3 – sqrt{3}}{5}) and (frac{3 + sqrt{3}}{5}).
- For what values of (k), ((4 – k)x^2 + (2k + 4)x + 8k + 1 = 0) is a perfect square?
- A polygon of (n) sides has (frac{n(n – 3)}{2}) diagonals. How many sides does a polygon with 54 diagonals have?
- Mr. Prakash was born in 1809 A.D. In year (x^2) A.D., he was (x – 3) years old. Find the value of (x).
- A rectangle of perimeter 34 units is inscribed in a circle of diameter 13 units. Find its sides.
- Solve the equation: (2(x – 3)^2 + 3(x – 2)(2x – 3) = 8(x + 4)(x – 4) – 1).
- Abhishek takes 6 days less than the time taken by Anubhav to finish a piece of work. If both of them together finish the work in 4 days, find the time taken by Anubhav alone to finish the work.
- Solve:
- (i) (frac{x – 1}{x – 2} + frac{x – 3}{x – 4} = frac{10}{3})
- (ii) (frac{1}{x + 1} + frac{2}{x + 2} = frac{4}{x + 4}), (x neq -1, 2)
- A two-digit number is four times the sum and three times the product of its digits. Find the number.
- Find the value of (k) for real and equal roots.
- (i) ((k + 1)x^2 – 2(k – 1)x + 1 = 0)
- (ii) (k^2 x^2 – 2(2k – 1)x + 4 = 0)
- If (alpha, beta) are the roots of the quadratic equation (x^2 + ax + b = 0), find the value of ((alpha^2 – 3)(beta^2 – 3)).
- If (alpha, beta) are the roots of (2x^2 + x + 7 = 0), find the value of:
- (i) (alpha^2 beta + alpha beta^2)
- (ii) (alpha^3 + beta^3 + 9alpha beta (alpha^2 + beta^2) + 3(alpha^2 beta + alpha beta^2))
- (iii) (alpha^4 + alpha^2 beta^2 + beta^4)
- If (alpha) and (beta) are the roots of the quadratic equation (x^2 – 6x + k = 0), find the value of (k) such that (alpha^2 + beta^2 = 40).
- In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increases by 30 minutes. Find the duration of the flight.
- Find two consecutive natural numbers whose product is 20.
- Find the whole number which, when decreased by 20, is equal to 69 times the reciprocal of the number.
- Find the value of (a) such that the quadratic equation ((a – 12)x^2 + 2(a – 12)x + 2 = 0) has equal roots.
- Two circles touch externally. The sum of their areas is (130pi , text{sq cm}) and the distance between their centers is 14 cm. Find the radii of the two circles.
- The speed of the boat in still water is 15 km/hr. It can go 30 km upstream and return downstream to the original point in 4 hrs. and 30 minutes. Find the speed of the stream.
- (ABCD) is a square. (F) is the midpoint of (AB). (BE) is one-third of (BC). If the area of (triangle AFB E) is 108 sq cm, find the length of (AC).
- Find the value of (sqrt{8 + 2sqrt{8 + 2sqrt{8 + ldots}}}).
- In a group of children, each child gives a gift to every other. If the number of gifts is 132, find the number of children.
- A sum of Rs.1200 becomes Rs.1333 in 2 years at compound interest compounded annually. Find the rate of interest.
- The roots (alpha) and (beta) of the quadratic equation (x^2 – 5x + 3(k – 1) = 0) are such that (alpha – beta = 1). Find (k).
- The difference of the squares of two numbers is 45. The square of the larger number is 4 times the square of the smaller number. Determine the numbers.
- If the equation ((1 + m^2)x^2 + 2mcx + (c^2 – a^2) = 0) has equal roots, prove that (c^2 = a^2 (1 + m^2)).
- Solve for (x):
[ x = frac{1}{1 – frac{1}{1 – frac{1}{2 – frac{1}{2 – x}}}} ] - Solve:
- (i) ( x^2 + left(frac{a + b}{a} + frac{a}{a + b}right)x + 1 = 0)
- (ii) ( frac{2x}{x – 3} + frac{x}{2x + 3} + frac{3x + 9}{(x – 3)(2x + 3)} = 0)
- The area of an isosceles triangle is 60 cm(^2) and the length of each one of its equal sides is 13 cm. Find its base.
- Solve: (3^{x+1} – 2 times 3^{x+2} = 81).
- Solve:
[ frac{1}{x + 1} + frac{1}{x + 5} = frac{1}{x + 2} + frac{1}{x + 4} ] - Solve the equations:
- (i) (sqrt{2x + sqrt{2x + 4}} = 4)
- (ii) (4x^2 – 4x^2 – 7x^2 – 4x + 4 = 0, , x neq 0)
- Solve for (x): (4x^2 – 2(a^2 + b^2)x + a^2 b^2 = 0).
- Solve for (x): (4x^2 – 4a^2 x + (a^4 – b^4) = 0).
- Solve for (x): (9x^2 – 9(a + b)x + [2a^2 + 5ab + 2b^2] = 0).
- Using the quadratic formula, solve the following quadratic equation for (x):
[ p^2 x^2 + (p^2 – q^2)x – q^2 = 0 ] - Using the quadratic formula, solve the following quadratic equation for (x):
[ x^2 – 2ax + (a^2 – b^2) = 0 ] - Using the quadratic formula, solve the following quadratic equation for (x):
[ x^2 – 4ax + 4a^2 – b^2 = 0 ] - Solve for (x): (9x^2 – 6a^2 x + (a^4 – b^4) = 0).
- Solve for (x): (9x^2 – 6ax + (a^2 – b^2) = 0).
- Solve for (x): (16x^2 – 8a^2 x + (a^4 – b^4) = 0).
- Solve for (x): (36x^2 – 12ax + (a^2 – b^2) = 0).
- If the roots of the equation ((a – b)x^2 + (b – c)x + (c – a) = 0) are equal, then:
- (A) (2b = a + c)
- (B) (2a = b + c)
- (C) (2c = a + b)
- (D) (frac{1}{b} = frac{1}{a} + frac{1}{c})
- If one of the roots of (x^2 + ax + 4 = 0) is twice the other root, then the value of (a) is:
- (A) (-3sqrt{2})
- (B) (8sqrt{2})
- (C) (sqrt{2})
- (D) (-2sqrt{2})
- In the equation (2x^2 – hx + 2k = 0), the sum of the roots is 4 and the product of the roots is (-3). Then (h) and (k) respectively, have the values:
- (A) 8 and 6
- (B) 4 and (-3)
- (C) (-3) and 8
- (D) 8 and (-3)
- (alpha) and (frac{1}{alpha}) are zeroes of the polynomial (4x^2 – 2x + (k – 4)). Find the value of (k).
- If (alpha, beta) are zeroes of (x^2 + 5x + 5), find the value of (alpha^{-1} + beta^{-1}).
- If (alpha, beta) are zeroes of (x^2 + 7x + 7), find the value of (frac{1}{alpha} + frac{1}{beta} – 2 alpha beta).
- Find the zeroes of (sqrt{3}x^2 + 10x + 7sqrt{3}).
- Write a quadratic polynomial whose one zero is (3 – sqrt{5}) and the product of zeroes is 4.
- Solve (x^{2/3} – 2x^{1/3} = 15).
- Solve the following by reducing to quadratic equations:
- (i) (x^4 – 26x^2 + 25 = 0)
- (ii) (2x – frac{3}{x} = 5)
- (iii) (sqrt{3x^2 – 2} = 2x – 1)
- (iv) (sqrt{3x + 10} + sqrt{6 – x} = 6)
- (v) (4left(x^2 + frac{1}{x^2}right) + 8left(x – frac{1}{x}right) – 3 = 0)
- (vi) (sigmaleft(x^2 + frac{1}{x^2}right) – 25left(x – frac{1}{x}right) + 12 = 0)
- (vii) (x^4 + 2x^3 – 13x^2 + 2x + 1 = 0)
- (viii) ((x – 7)(x – 3)(x + 1)(x + 5) – 1680 = 0)
- Solve (frac{1}{a + b + x} = frac{1}{a} + frac{1}{b} + frac{1}{x}), ((a + b) neq 0).
- If (-4) is a root of the quadratic equation (x^2 + kx – 4 = 0) and the quadratic equation (x^2 + px + k = 0) has equal roots, find the values of (k) and (p).
- If (x = 2) and (x = 3) are the roots of the equation (3x^2 – 2kx + 2m = 0), find the values of (k) and (m).