Question

solve this plz
fast as fast step by step​

Answer 1

[1]

We have to find the values of k for which the the quadratic equation x² - 2kx + 7k - 12 = 0

Comparing the given quadratic equation with the standard form of quadratic equation i.e., ax² + bx + c = 0 , we have

• a = 1
• b = -2k
• c = 7k - 12

We know, A quadratic equation will have equal roots only when the discriminant of the given quadratic equation is 0. , So we have

⇒ D = 0

⇒ b² - 4ac = 0

⇒ b² = 4ac

⇒ (- 2k)² = 4(1)(7k - 12)

⇒ 4k² = 28k - 48

Divide both sides by 4,

⇒ k² - 7k + 12 = 0

⇒ k² - 4k - 3k + 12 = 0

⇒ k(k - 4) - 3(k - 4) = 0

⇒ (k - 4)(k - 3) = 0

∴ k = 4, 3

[2]

We have,

⇒ x - 18/x = 6

⇒ x² - 18 = 6x

⇒ x² - 6x - 18 = 0

Using the quadratic formula, we have

⇒ x = { - b ± √(b² - 4ac) } / { 2a }

⇒ x = ( 6 ± √108 ) / 2

⇒ x = (6 ± 6√3) / 2

⇒ x = 3 ± 3√3

We have two cases now,

Case 1 ( Solving with positive sign )

⇒ x = 3 + 3√3

⇒ x = 3(1 + √3)

⇒ x = 3(1 + 1.73) [ √3 = 1.73 ]

⇒ x = 3 × 2.73

⇒ x = 8.19

⇒ x = 8.2 [ 2 Significant figures ]

Case 2 ( Solving with negative sign )

⇒ x = 3 - 3√3

⇒ x = 3(1 - √3)

⇒ x = 3(1 - 1.73) [ √3 = 1.73 ]

⇒ x = 3 × -0.73

⇒ x = -2.19

⇒ x = -2.2 [ 2 Significant figures ]

Finally, The values of x are 8.2 and -2.2 correct to two significant figures.

[3]

Given quadratic equation,

▪ 3x² - 4√3x + 4 = 0

The nature of the roots of a quadratic equation can be defined by the discriminant of that Quadratic equation, which is equal to b² - 4ac

where,

• b = coefficient of x
• a = coefficient of x²
• c = constant term

So, Let us find the value of the discriminant,

⇒ D = b² - 4ac

⇒ D = (-4√3)² - 4×3×4

⇒ D = 48 - 48

⇒ D = 0

When the discriminant of a quadratic equation is equal to zero, real and equal roots exist for that Quadratic equation.

Here, The roots are real and equal.

[4]

Again, we have to find the value of a constant in a quadratic equation for which the quadratic equation will have equal and real roots, Same as 1.

We have,

▪ (3m + 1)x² + 2(m + 1)x + m = 0

We have to find the values of m, for which the equation will have real and equal roots. clearly the discriminant will be zero then, so we have

⇒ D = b² - 4ac

• b = Coefficient of x
• a = Coefficient of x²
• c = Constant term

⇒ D = 0 [ for real and equal roots ]

⇒ (2m + 2)² - 4(3m + 1)(m) = 0

⇒ 4m² + 4 + 8m = 12m² + 4m

⇒ 8m² - 4m - 4 = 0

Divide both sides by 4,

⇒ 2m² - m - 1 = 0

⇒ 2m² - 2m + m - 1 = 0

⇒ 2m(m - 1) + 1(m - 1) = 0

⇒ (m - 1)(2m + 1) = 0

∴ m = 1 , -1/2

[5]

Given that the Difference of squares of two numbers is 45.

Also, the square of the smaller number is 4 times the larger number.

Let the numbers be x and y respectively. x > y

Case 1 :-

▪ Difference of squares is 45.

⇒ x² - y² = 45 ...(1)

Case 2 :-

▪ Square of the smaller number is 4 times the larger number.

⇒ y² = 4x ...(2)

Substituting value of y² from (2) in (1), we get

⇒ x² - 4x = 45

⇒ x² - 4x - 45 = 0

⇒ x² - 9x + 5x - 45 = 0

⇒ x(x - 9) + 5(x - 9) = 0

⇒ (x - 9)(x + 5) = 0

∴ x = 9, -5

Case 1 ( x = 9 )

Substituting x = 9 in (2), we have

⇒ y² = 4×9

⇒ y² = 36

⇒ y = ± 6

Case 2 ( x = -5 )

Substituting x = -5 in (2), we have

⇒ y² = 4×-5

⇒ y² = -20

Here, It y is an imaginary number, so we neglect the value of x = -5.

So, The numbers can be 9 and 6 or 9 and -6.