2x2 – 7x + 3 = 0 Dividing both side by 2 x² - 7/2 x + 3/2 = 0 ⇒ x² - 7/2 x = - 3/2 Adding [1/2(7/2)]² on both the sides, we get x² - 7/2 x + (7/4)² = - 3/2 + (7/4)² [(as x = -b±√(b²- 4ac)]/2a] ⇒ (x - 7/4)² = - 3/2 + 49/16 ⇒ (x - 7/4)² = - 24 + 49/16 ⇒ (x - 7/4)² = 25/16 ⇒ x - 7/4 = ± 5/4 Either x -Read more

2×2 – 7x + 3 = 0
Dividing both side by 2
x² – 7/2 x + 3/2 = 0 ⇒ x² – 7/2 x = – 3/2
Adding [1/2(7/2)]² on both the sides, we get
x² – 7/2 x + (7/4)² = – 3/2 + (7/4)² [(as x = -b±√(b²- 4ac)]/2a]
⇒ (x – 7/4)² = – 3/2 + 49/16 ⇒ (x – 7/4)² = – 24 + 49/16
⇒ (x – 7/4)² = 25/16
⇒ x – 7/4 = ± 5/4
Either x – 7/4 = 5/4 or x – 7/4 = – 5/4
⇒ x = 5/4 + 7/4 or x – 5/4 + 7/4
⇒ x = 5+7/4 = 12/4 = 3 x 5+7/4 = 2/4 = 1/2
Hence, the roots of the quadratic equation are 3 and 1/2.

x2 – 2x = (–2) (3 – x) simplifying the given equation, we get x2 – 2x = (–2) (3 – x) ⇒ x² - 2x = - 6 + 2x ⇒ x² - 4x + 6 = 0 or x² - 4x + 6 = 0 This is an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

x2 – 2x = (–2) (3 – x)
simplifying the given equation, we get
x2 – 2x = (–2) (3 – x)
⇒ x² – 2x = – 6 + 2x
⇒ x² – 4x + 6 = 0
or x² – 4x + 6 = 0
This is an equation of type ax² + bx + c = 0.
Hence, the given equation is a quadratic equation.

(x – 2)(x + 1) = (x – 1)(x + 3) Simplifying the given equation, we get (x – 2)(x + 1) = (x – 1)(x + 3) ⇒ x² - 2x + x - 2 = x² - x +3x - 3 ⇒ - 3x + 1 = 0 or 3x - 1 = 0 This is not an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

(x – 2)(x + 1) = (x – 1)(x + 3)
Simplifying the given equation, we get
(x – 2)(x + 1) = (x – 1)(x + 3)
⇒ x² – 2x + x – 2 = x² – x +3x – 3
⇒ – 3x + 1 = 0
or 3x – 1 = 0
This is not an equation of type ax² + bx + c = 0.
Hence, the given equation is a quadratic equation.

x² + 3x + 1 = (x – 2)² simplifying the given equation, we get x² + 3x + 1 = (x – 2)² ⇒ x² + 3x + 1 = x² - 4x + 4 ⇒ 7x - 3 = 0 or 7x - 3 = 0 This is not an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

x² + 3x + 1 = (x – 2)²
simplifying the given equation, we get
x² + 3x + 1 = (x – 2)²
⇒ x² + 3x + 1 = x² – 4x + 4
⇒ 7x – 3 = 0
or 7x – 3 = 0
This is not an equation of type ax² + bx + c = 0.
Hence, the given equation is a quadratic equation.

(2x – 1)(x – 3) = (x + 5)(x – 1) Simplifying the given equation, we get (2x – 1)(x – 3) = (x + 5)(x – 1) ⇒ 2x² - x - 6x + 3 = x² + 5x - x - 5 ⇒ x² - 11x + 8 = 0 or x² - 11x + 8 = 0 This is an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

(2x – 1)(x – 3) = (x + 5)(x – 1)
Simplifying the given equation, we get
(2x – 1)(x – 3) = (x + 5)(x – 1)
⇒ 2x² – x – 6x + 3 = x² + 5x – x – 5
⇒ x² – 11x + 8 = 0
or x² – 11x + 8 = 0
This is an equation of type ax² + bx + c = 0.
Hence, the given equation is a quadratic equation.

(x – 3)(2x +1) = x(x + 5) Simplifying the given equation, we get (x – 3)(2x +1) = x(x + 5) ⇒ 2x² - 6x + x - 3 = x² + 5x ⇒ x² - 10x - 3 = 0 or x² - 10x - 3 = 0 This is an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

(x – 3)(2x +1) = x(x + 5)
Simplifying the given equation, we get
(x – 3)(2x +1) = x(x + 5)
⇒ 2x² – 6x + x – 3 = x² + 5x
⇒ x² – 10x – 3 = 0
or x² – 10x – 3 = 0
This is an equation of type ax² + bx + c = 0.
Hence, the given equation is a quadratic equation.

Let, the breadth of plot = x m Therefore, the length of plot 2x + 1 m Hence, area = x(2x + 1) m² According to the questions, x(2x + 1) = 528 ⇒ 2x² + x = 528 ⇒ 2x² + x - 528 = 0 Hence, the length and breath of the plot satisfies the equation 2x² + x - 528 = 0.

Let, the breadth of plot = x m
Therefore, the length of plot 2x + 1 m
Hence, area = x(2x + 1) m²
According to the questions, x(2x + 1) = 528
⇒ 2x² + x = 528
⇒ 2x² + x – 528 = 0
Hence, the length and breath of the plot satisfies the equation 2x² + x – 528 = 0.

(x + 2)³ = 2x (x² – 1) Simplifying the given equation, we get (x + 2)³ = 2x (x² – 1) ⇒ x³ + 6x² + 12x + 8 = 2x³ - 2x ⇒ - x³ + 6x² + 14x + 8 = 0 or x³ - 6x² - 14x - 8 = 0 This is not an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

(x + 2)³ = 2x (x² – 1)
Simplifying the given equation, we get
(x + 2)³ = 2x (x² – 1)
⇒ x³ + 6x² + 12x + 8 = 2x³ – 2x
⇒ – x³ + 6x² + 14x + 8 = 0
or x³ – 6x² – 14x – 8 = 0
This is not an equation of type ax² + bx + c = 0.
Hence, the given equation is a quadratic equation.

Let the first integer = x therefore, the second integer = x + 1 Hence, the product = x(x + 1) According to the questions, x(x + 1) = 306 ⇒ x² + x = 306 ⇒ x² + x - 306 = 0 Hence, the two consecutive integers satisfies quadratic equation x² + x - 306 = 0

Let the first integer = x
therefore, the second integer = x + 1
Hence, the product = x(x + 1)
According to the questions, x(x + 1) = 306
⇒ x² + x = 306
⇒ x² + x – 306 = 0
Hence, the two consecutive integers satisfies quadratic equation x² + x – 306 = 0

## Find the roots of the following quadratic equations, if they exist, by the method of completing the square: 2 x² – 7x + 3 = 0

2x2 – 7x + 3 = 0 Dividing both side by 2 x² - 7/2 x + 3/2 = 0 ⇒ x² - 7/2 x = - 3/2 Adding [1/2(7/2)]² on both the sides, we get x² - 7/2 x + (7/4)² = - 3/2 + (7/4)² [(as x = -b±√(b²- 4ac)]/2a] ⇒ (x - 7/4)² = - 3/2 + 49/16 ⇒ (x - 7/4)² = - 24 + 49/16 ⇒ (x - 7/4)² = 25/16 ⇒ x - 7/4 = ± 5/4 Either x -Read more

2×2 – 7x + 3 = 0

See lessDividing both side by 2

x² – 7/2 x + 3/2 = 0 ⇒ x² – 7/2 x = – 3/2

Adding [1/2(7/2)]² on both the sides, we get

x² – 7/2 x + (7/4)² = – 3/2 + (7/4)² [(as x = -b±√(b²- 4ac)]/2a]

⇒ (x – 7/4)² = – 3/2 + 49/16 ⇒ (x – 7/4)² = – 24 + 49/16

⇒ (x – 7/4)² = 25/16

⇒ x – 7/4 = ± 5/4

Either x – 7/4 = 5/4 or x – 7/4 = – 5/4

⇒ x = 5/4 + 7/4 or x – 5/4 + 7/4

⇒ x = 5+7/4 = 12/4 = 3 x 5+7/4 = 2/4 = 1/2

Hence, the roots of the quadratic equation are 3 and 1/2.

## Check whether the following is quadratic equations : x² – 2x = (–2) (3 – x)

x2 – 2x = (–2) (3 – x) simplifying the given equation, we get x2 – 2x = (–2) (3 – x) ⇒ x² - 2x = - 6 + 2x ⇒ x² - 4x + 6 = 0 or x² - 4x + 6 = 0 This is an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

x2 – 2x = (–2) (3 – x)

See lesssimplifying the given equation, we get

x2 – 2x = (–2) (3 – x)

⇒ x² – 2x = – 6 + 2x

⇒ x² – 4x + 6 = 0

or x² – 4x + 6 = 0

This is an equation of type ax² + bx + c = 0.

Hence, the given equation is a quadratic equation.

## Check whether the following is quadratic equation: (x – 2)(x + 1) = (x – 1)(x + 3)

(x – 2)(x + 1) = (x – 1)(x + 3) Simplifying the given equation, we get (x – 2)(x + 1) = (x – 1)(x + 3) ⇒ x² - 2x + x - 2 = x² - x +3x - 3 ⇒ - 3x + 1 = 0 or 3x - 1 = 0 This is not an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

(x – 2)(x + 1) = (x – 1)(x + 3)

Simplifying the given equation, we get

(x – 2)(x + 1) = (x – 1)(x + 3)

⇒ x² – 2x + x – 2 = x² – x +3x – 3

⇒ – 3x + 1 = 0

or 3x – 1 = 0

This is not an equation of type ax² + bx + c = 0.

Hence, the given equation is a quadratic equation.

## Check whether the following is quadratic equation: x² + 3x + 1 = (x – 2)²

x² + 3x + 1 = (x – 2)² simplifying the given equation, we get x² + 3x + 1 = (x – 2)² ⇒ x² + 3x + 1 = x² - 4x + 4 ⇒ 7x - 3 = 0 or 7x - 3 = 0 This is not an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

x² + 3x + 1 = (x – 2)²

See lesssimplifying the given equation, we get

x² + 3x + 1 = (x – 2)²

⇒ x² + 3x + 1 = x² – 4x + 4

⇒ 7x – 3 = 0

or 7x – 3 = 0

This is not an equation of type ax² + bx + c = 0.

Hence, the given equation is a quadratic equation.

## Check whether the following is quadratic equation: (2x – 1)(x – 3) = (x + 5)(x – 1)

(2x – 1)(x – 3) = (x + 5)(x – 1) Simplifying the given equation, we get (2x – 1)(x – 3) = (x + 5)(x – 1) ⇒ 2x² - x - 6x + 3 = x² + 5x - x - 5 ⇒ x² - 11x + 8 = 0 or x² - 11x + 8 = 0 This is an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

(2x – 1)(x – 3) = (x + 5)(x – 1)

See lessSimplifying the given equation, we get

(2x – 1)(x – 3) = (x + 5)(x – 1)

⇒ 2x² – x – 6x + 3 = x² + 5x – x – 5

⇒ x² – 11x + 8 = 0

or x² – 11x + 8 = 0

This is an equation of type ax² + bx + c = 0.

Hence, the given equation is a quadratic equation.

## Check whether the following is quadratic equation: (x – 3)(2x +1) = x(x + 5)

(x – 3)(2x +1) = x(x + 5) Simplifying the given equation, we get (x – 3)(2x +1) = x(x + 5) ⇒ 2x² - 6x + x - 3 = x² + 5x ⇒ x² - 10x - 3 = 0 or x² - 10x - 3 = 0 This is an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

(x – 3)(2x +1) = x(x + 5)

Simplifying the given equation, we get

(x – 3)(2x +1) = x(x + 5)

⇒ 2x² – 6x + x – 3 = x² + 5x

⇒ x² – 10x – 3 = 0

or x² – 10x – 3 = 0

This is an equation of type ax² + bx + c = 0.

Hence, the given equation is a quadratic equation.

## Represent the situations in the form of quadratic equations : The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Let, the breadth of plot = x m Therefore, the length of plot 2x + 1 m Hence, area = x(2x + 1) m² According to the questions, x(2x + 1) = 528 ⇒ 2x² + x = 528 ⇒ 2x² + x - 528 = 0 Hence, the length and breath of the plot satisfies the equation 2x² + x - 528 = 0.

Let, the breadth of plot = x m

Therefore, the length of plot 2x + 1 m

Hence, area = x(2x + 1) m²

According to the questions, x(2x + 1) = 528

⇒ 2x² + x = 528

⇒ 2x² + x – 528 = 0

Hence, the length and breath of the plot satisfies the equation 2x² + x – 528 = 0.

## Check whether the following is quadratic equations : (x + 2)³ = 2x (x² – 1)

(x + 2)³ = 2x (x² – 1) Simplifying the given equation, we get (x + 2)³ = 2x (x² – 1) ⇒ x³ + 6x² + 12x + 8 = 2x³ - 2x ⇒ - x³ + 6x² + 14x + 8 = 0 or x³ - 6x² - 14x - 8 = 0 This is not an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

(x + 2)³ = 2x (x² – 1)

See lessSimplifying the given equation, we get

(x + 2)³ = 2x (x² – 1)

⇒ x³ + 6x² + 12x + 8 = 2x³ – 2x

⇒ – x³ + 6x² + 14x + 8 = 0

or x³ – 6x² – 14x – 8 = 0

This is not an equation of type ax² + bx + c = 0.

Hence, the given equation is a quadratic equation.

## Represent the situations in the form of quadratic equations : The product of two consecutive positive integers is 306. We need to find the integers.

Let the first integer = x therefore, the second integer = x + 1 Hence, the product = x(x + 1) According to the questions, x(x + 1) = 306 ⇒ x² + x = 306 ⇒ x² + x - 306 = 0 Hence, the two consecutive integers satisfies quadratic equation x² + x - 306 = 0

Let the first integer = x

See lesstherefore, the second integer = x + 1

Hence, the product = x(x + 1)

According to the questions, x(x + 1) = 306

⇒ x² + x = 306

⇒ x² + x – 306 = 0

Hence, the two consecutive integers satisfies quadratic equation x² + x – 306 = 0

## Find the roots of the following quadratic equations by factorisation: 2x² + x – 6 = 0

2x² + x – 6 = 0 Solving the quadratic equation, we get 2x² + x – 6 = 0 ⇒ 2x² - 4x + 3x - 6 = 0 ⇒ 2x(x - 2) +3 (x - 2) = 0 ⇒ (x - 2) (2x + 3) = 0 ⇒ (x - 2) = 0 or (2x + 3) = 0 Either x = 2 or x = - 3/2 Hence, the roots of the given quadratic equation are 2 and - 3/2.

2x² + x – 6 = 0

See lessSolving the quadratic equation, we get

2x² + x – 6 = 0 ⇒ 2x² – 4x + 3x – 6 = 0

⇒ 2x(x – 2) +3 (x – 2) = 0

⇒ (x – 2) (2x + 3) = 0

⇒ (x – 2) = 0 or (2x + 3) = 0

Either x = 2 or x = – 3/2

Hence, the roots of the given quadratic equation are 2 and – 3/2.