Solving Equations Worksheet1 Solution8





In this page solving equations worksheet1 solution8 we are going to see solution of practice questions.

Question 8:

Solving equation √(x² - 9 x + 18) + √(x² + 2 x - 15) = √(x² - 4 x + 3) following roots are obtained

Solution:

√(x² - 9 x + 18) + √(x² + 2 x - 15) = √(x² - 4 x + 3)

For removing the square root on left side first we have to take square on both sides 

[√(x² - 9 x + 18) + √(x² + 2 x - 15)]² = [√(x² - 4 x + 3)]²

[√(x²-9x+18)]²+[√(x²+2x-15)]²+2√(x²-9x+18)(x²+2x-15)= (x²-4x+3)

x²-9x+18+x²+2x-15 + 2 √(x²-9x+18)(x²+2x-15) = (x²-4x+3)

2 x²-7 x + 3 + 2√(x²-9x+18)(x²+2x-15) = x²-4x+3

2√(x²-9x+18)(x²+2x-15) = x² - 4 x + 3 - 2 x² + 7 x - 3

2√(x²-9x+18)(x²+2x-15) = - x² + 3 x

now we are going to take squares on both sides

[2√(x²-9x+18)(x²+2x-15)]² = [- x² + 3 x]²

4(x²-9x+18)(x²+2x-15) = x⁴ + 2 (-x²)(3 x) + (3 x)²

4[x⁴+2x³-15x²-9x³-18x²+135x+18x²+36x-270] = x⁴ - 6 x³ + 9x²

4 [x⁴ - 7x³ -15 x² + 171 x - 270] = x⁴ - 6 x³ + 9x²

4x⁴ - 28 x³ -60 x² + 684 x - 1080 = x⁴ - 6 x³ + 9x²

4x⁴ - x⁴ - 28 x³ + 6 x³ - 60 x² - 9x²+ 684 x - 1080 = 0

3 x⁴ - 22 x³ - 69 x² + 684 x - 1080 = 0

By using synthetic division we got two factors (x-3) (x -3). By solving this we get two roots.To find other two roots we have to solve the quadratic equation 3 x² - 4 x - 120 = 0.

We cannot solve this equation by using factorization method. So let us use formula.

a = 3   b = -4 and c = -120

  x = [- b ± √(b² - 4 ac)]/2a

  x = [-(-4) ± √(-4)² - 4 (3)(-120))]/2(3)

  x = [4 ± √(16 + 1440)]/6

  x = [4 ± √1456]/6

  x = [4 ± √2 x 2 x 2 x 2 x 91]/6

  x = [4 ± 4 √91]/6

  x = 2[2 ± 2 √91]/6

  x = [2 ± 2 √91]/3

x - 3 = 0

   x = 3

roots are x = 3 , 3 , [2 ± 2 √91]/3

(A)   3 , 1 , [2 ± 2 √94]/3

(B)    3 , -3 , [2 ± 2 √91]/3

(C)   3 , 3 , [2 ± 2 √91]/3

(D)   5, 3 , [2 ± 2 √91]/3

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