第 10 类 RD Sharma 解——第 8 章二次方程——练习 8.1

问题 1. 以下哪些是二次方程？

(i) x ² + 6x – 4 = 0

解决方案：

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is x² + 6x – 4 =0  in quadratic equation.

So, It is a quadratic equations.

(ii) √3x ² − 2x + ½= 0

解决方案：

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is √3x² − 2x + ½= 0 in quadratic equation.

So, It is a quadratic equation.

(iii) x ² + 1/x ² = 5

解决方案：

This equation can be written as: x⁴ – 5x² + 1 =0

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is x⁴ – 5x² + 1 =0 is 4 degree polynomial.

So, It is not quadratic equations.

(iv) x – 3/x = x ²

解决方案：

This equation can be written as: x³ – x² – 3 = 0

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is x³ – x² – 3 = 0 is 3 degree polynomial.

So, It is not quadratic equations.

(v) 2x ² – √3x + 9 = 0

解决方案：

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is 2x² – √3x + 9 = 0  in quadratic equation.

So, It is a quadratic equation.

(vi) x ² – 2x -√x – 5 = 0

解决方案：

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is x² – 2x -√x -5 = 0.

So, It is not quadratic equation.

(vii) 3x ² - 5x + 9 = x ² - 7x + 3

解决方案：

This equation can be written as: 2x² + 2x + 6 = 0

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is 2x² + 2x + 6 = 0 in quadratic equation.

So, It is a quadratic equation.

(八) x + 1/x = 1

解决方案：

This equation can be written as: x² – x + 1 = 0

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is x² – x + 1 = 0 in quadratic equation.

So, It is a quadratic equation.

(ix) x ² – 3x = 0

解决方案：

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is  x² – 3x =0  in quadratic equation.

So, It is a quadratic equation.

(x) (x + 1/x) ² = 3(x +1/x)

解决方案：

((x² + 1)/x)² = 3((x² + 1)/2)

This equation can be written as: x⁵ – 3x⁴ + 2x³ – 3x² + x =0

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is x⁵ – 3x⁴ + 2x³ – 3x² + x = 0 is 5 degree polynomial.

So, It is not quadratic equations.

(xi) (2x ² + 1)(3x ² + 2) = 6(x ² - 1)(x ² - 2)

解决方案：

This equation can be written as:

6x² + 4x + 3x + 2 = 6x² -12x – 6x + 12

7x + 2 = -18x + 12

This equation can be written as: 25x – 10 = 0 

As we know that , quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is 25x – 10 = 0 is 1 degree polynomial.

So, It is not quadratic equations.

(xii) x + 1/x = x ² , x ≠ 0

解决方案：

x + 1/x = x²

On multiplying by x on both sides we have,
x² + 1 = x³

This equation can be written as: x³ – x² – 1 = 0

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation x³ – x² – 1 = 0 is  = 0 is 3 degree polynomial.

So, It is not quadratic equations.

(xiii) 16x ² – 3 = (2x + 5)(5x – 3)

解决方案：

16x² – 3 = (2x + 5)(5x – 3)

16x² – 3 = 10x² – 6x + 25x – 15

This equation can be written as: 6x² – 19x + 12 = 0

As we know that , quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is 6x² – 19x + 12 = 0 is 2 degree polynomial.

So, It is quadratic equations.

(xiv) (x + 2) ³ = x ³ – 4

解决方案：

(x + 2)³ = x³ – 4

On expanding, we get
x³ + 6x² + 8x + 8 = x³ – 4

6x² + 8x + 12 = 0

This equation can be written as: 6x² + 8x + 12 = 0

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is 6x² – 19x + 12 = 0 is 2 degree polynomial.

So, It is quadratic equations.

(xv) x(x + 1) + 8 = (x + 2)(x – 2)

解决方案：

x(x + 1) + 8 = (x + 2)(x – 2)

x² + x + 8 = x² – 4

x + 12 = 0 

This equation can be written as: x + 12 = 0 

As we know that, quadratic equation is in form of: ax² + bx + c = 0

Here, given equation is x + 12 = 0 is 0 degree polynomial.

So, It is not a quadratic equation.

问题 2. 在以下每一项中，确定给定值是否是给定方程的解：

(i) x ² – 3x + 2 = 0, x = 2, x = – 1

解决方案：

Here we have check values to determine the solution the given equation:

LHS = x²– 3x + 2

Now we will put x = 2 in LHS, we get

(2)² – 3(2) + 2

⇒ 4 – 6 + 2 = 0 = RHS

⇒ LHS = RHS

So, we can say that x = 2 is not a solution of the given equation.

Similarly,

Now, we will put x = -1 in LHS, we get

(-1)² – 3(-1) + 2

1 + 3 + 2 = 6 ≠ RHS

LHS ≠ RHS

So, we can say that x = -1 is not a solution of the given equation.

(ii) x ² + x + 1 = 0, x = 0, x = 1

解决方案：

Here we have check values to determine the solution the given equation:

LHS = x² + x + 1

Now we will put x = 0 in LHS, we get

(0)² + 0 + 1

⇒ 1 ≠ RHS

⇒ LHS ≠ RHS

So , we can say that x = 0 is not a solution of the given equation.

Similarly,

Now, we will put x = 1 in LHS, we get

(1)² + 1 + 1

⇒ 3 ≠ RHS

⇒ LHS ≠ RHS

So, we can say that x = 1 is not a solution of the given equation.

(iii) x ² - 3√3x + 6 = 0, x = √3 和 x = -2√3

解决方案：

Here we have check values to determine the solution the given equation:

LHS = x² − 3√3x + 6

Now, we will put x = √3 in LHS, we get

(√3)² − 3√3(√3) + 6

⇒ 3 – 9 + 6 = 0 = RHS

⇒ LHS = RHS

So , we can say that x = √3 is a solution of the given equation.

Similarly,

Now, we will put x = −2√3 in LHS, we get

(-2√3)² − 3√3(-2√3) + 6

⇒ 12 + 18 + 6 = 36 ≠ RHS

⇒ LHS ≠ RHS

So, we can say that x = −2√3 is not a solution of the given equation.

(iv) x + 1/x = 13/6, x = 5/6, x = 4/3

解决方案：

Here we have check values to determine the solution the given equation:

LHS = x +1/ x

Now, we will put x = 5/6 in LHS, we get

(5/6) + 1/(5/6) = 5/6 + 6/5

⇒ 61/30 ≠ RHS

⇒ LHS ≠ RHS

So , we can say that x = 5/6 is not a solution of the given equation.

Similarly,

Now, we will put x = 4/3 in LHS, we get

(4/3) + 1/(4/3) = 4/3 + 3/4

⇒ 25/12 ≠ RHS

⇒ LHS ≠ RHS

So, we can say that x = 3/4 is not a solution of the given equation.

(v) 2x ² – x + 9 = x ² + 4x + 3, x = 2, x = 3

解决方案：

Here we have check values to determine the solution the given equation:

2x²– x + 9 = x² + 4x + 3

⇒ x² – 5x + 6 = 0

LHS = x²– 5x + 6

Now, we will put x = 2 in LHS, we get

(2)² – 5(2) + 6

4 – 10 + 6 = 0 = RHS

⇒ LHS = RHS

So , we can say that x = 2 is a solution of the given equation.

Similarly,

Now, we will put x = 3 in LHS, we get

(3)² – 5(3) + 6

⇒ 9 – 15 + 6 = 0 = RHS

⇒ LHS = RHS

So, we can say that x = 3 is a solution of the given equation.

(vi) x ² – √2x – 4 = 0, x = -√2, x = -2√2

解决方案：

Here we have check values to determine the solution the given equation:

LHS = x² – √2x – 4

Now, we will put x = -√2 in LHS, we get

(-√2)² − √2(-√2) – 4

⇒ 4 + 2 – 4 = 2 ≠ RHS

⇒ LHS ≠ RHS

So , we can say that x = -√2 is a solution of the given equation.

Similarly,

Now, we will put x = −2√2 in LHS, we get

(-2√2)² − √2(-2√2) – 4

⇒ 8 + 4 – 4 = 8 ≠ RHS

⇒ LHS ≠ RHS

So, we can say that x = −2√2 is not a solution of the given equation.

(vii) a ² x ² – 3abx + 2b ² = 0, x = a/b, x = b/a

解决方案：

Here we have check values to determine the solution the given equation:

LHS =  a²x² – 3abx + 2b²

Now, we will put x = a/b in LHS, we get

a²(a/b)² -3ab(a/b) + 2b²

⇒ a⁴/b² – 3a² + 2b² ≠ RHS

⇒ LHS ≠ RHS

So, we can say that x = a/b is a solution of the given equation.

Similarly,

Now, we will put x = b/a in LHS, we get

a²(b/a)² – 3ab(b/a) + 2b²

⇒ b² – 3b² + 2b² = RHS

⇒ LHS = RHS

So, we can say that x = b/a is not a solution of the given equation.

问题 3：在以下每一项中，找出给定值是给定方程的解的 k 值。

(i) 7x ² + kx – 3 = 0, x = 2/3

解决方案：

Here we have to find value of k. So, we will put value of x = 2/3 in the given equation.

Here, x = 2/3

7(2/3)² + k(2/3) – 3 =0

28/9 + 2k/3 -3 =0

2k/3 = 3 – 28/9

2k/3 = (27 – 28)/9

2k/3 = -1/9

k = -1/6

On putting value of x = 2/3 , we will get k =-1/6.

(ii) x ² – x(a + b) + k = 0, x = a

解决方案：

Here we have to find value of k. So, we will put value of x = a in the given equation.

Here, x = a

a² – a² – ab + k = 0

k = ab

On putting value of x = a, we get k = ab.

(iii) kx ² + √2x – 4 = 0, x = √2

解决方案：

Here we have to find value of k. So, we will put value of x = √2 in the given equation.

Here, x = √2

k(√2)² + (√2)² – 4 = 0

2k + 2 – 4 = 0

2k – 2 = 0

k = 1

On putting value of x = √2 , we get k = 1.

(iv) x ² + 3ax + k = 0, x = -a

解决方案：

Here we have to find value of k. So, we will put value of x = -a in the given equation.

Here, x = -a

(- a)² + 3a(- a) + k = 0

a² – 3a² + k = 0

k = 2a²

On putting value of x = – a, we get k = 2a².

问题 4. 给定检查 3 是否是方程的根

√(x ² – 4x +3) + √(x ² -9) = √(4x ² – 14x + 16)

解决方案：

Here we have to check, 3 is root of equation or not.

So, we first put x = 3 in LHS side.

√((3)² – 4(3) + 3) + √((3)² – 9)

⇒ 0 + 0 = 0

Now , we will put value x = 3 in RHS side.

√(4(3)² – 14(3) + 16)

⇒ √(52 – 42) = √10

Now we can say that, LHS ≠ RHS.

So, x = 3 is not the root of the equation.

问题 5. 如果 x = 2/3 和 x = -3 是方程 ax ² + 7x + b = 0 的根，求 a 和 b 的值。

解决方案：

Here we have to find value of a and b, so we will put values x = 2/3 and x = -3 in the equation.

So first put x = 2/3

a(2/3)² + 7(2/3) + b =0

4a/9 + 14/3 + b =0

4a + 42 + 3b = 0     — (i)

Now, x = -3.

a(-3)² + 7(-3) + b = 0

9a – 21 + b = 0       –(ii)

Now solving these equations (i) & (ii).

After solving these equations, we get a = 3 and b = -6.