Solve the following using identities (Part-4)
1. 4 x^2 + 4 y^2 + 4 z^2 + 8 x y + 8 y z + 8 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }
4 x^2 + 4 y^2 + 4 z^2 + 8 x y + 8 y z + 8 z x
(2 x)^2 + (2 y)^2 + (2 z)^2 + 2 * 2 x * 2 y + 2 * 2 y * 2 z + 2 * 2 z * 2 x
(2 x + 2 y + 2 z)^2 (2 x + 2 y + 2 z) (2 x + 2 y + 2 z)
2. 9 x^2 + 9 y^2 + 9 z^2 + 18 x y + 18 y z + 18 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }9 x^2 + 9 y^2 + 9 z^2 + 18 x y + 18 y z + 18 z x
(3 x)^2 + (3 y)^2 + (3 z)^2 + 2 * 3 x * 3 y + 2 * 3 y * 3 z + 2 * 3 z * 3 x
(3 x + 3 y + 3 z)^2
(3 x + 3 y + 3 z) (3 x + 3 y + 3 z)
3. x^2 + 4 y^2 + 9 z^2 + 4 x y + 12 y z + 6 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }x^2 + 4 y^2 + 9 z^2 + 4 x y + 12 y z + 6 z x
(x)^2 + (2 y)^2 + (3 z)^2 + 2 * x * 2 y + 2 * 2 y * 3 z + 2 * 3 z * x
(x + 2 y + 3 z)^2
(x + 2 y + 3 z) (x + 2 y + 3 z)
4. 4 x^2 + 9 y^2 + z^2 + 12 x y + 6 y z + 4 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }4 x^2 + 9 y^2 + z^2 + 12 x y + 6 y z + 4 z x
(2 x)^2 + (3 y)^2 + (z)^2 + 2 * 2 x * 3 y + 2 * 3 y * z + 2 * z * 2 x
(2 x + 3 y + z)^2
(2 x + 3 y + z) (2 x + 3 y + z)
5. 9 x^2 + 4 y^2 + z^2 + 12 x y + 4 y z + 6 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }9 x^2 + 4 y^2 + z^2 + 12 x y + 4 y z + 6 z x
(3 x)^2 + (2 y)^2 + (z)^2 + 2 * 3 x * 2 y + 2 * 2 y * z + 2 * z * 3 x
(3 x + 2 y + z)^2
(3 x + 2 y + z) (3 x + 2 y + z)
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| maths-ncert-solution |
6. x^2 + 4 y^2 + 49 z^2 + 4 x y + 28 y z + 14 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }x^2 + 4 y^2 + 49 z^2 + 4 x y + 28 y z + 14 z x
(x)^2 + (2 y)^2 + (7 z)^2 + 2 * x * 2 y + 2 * 2 y * 7 z + 2 * 7 z * x
(x + 2 y + 7 z)^2
(x + 2 y + 7 z) (x + 2 y + 7 z)
7. 4 x^2 + 4 y^2 + 25 z^2 + 8 x y + 20 y z + 20 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }4 x^2 + 4 y^2 + 25 z^2 + 8 x y + 20 y z + 20 z x
(2 x)^2 + (2 y)^2 + (5 z)^2 + 2 * 2 x * 2 y + 2 * 2 y * 5 z + 2 * 5 z * 2 x
(2 x + 2 y + 5 z)^2
(2 x + 2 y + 5 z) (2 x + 2 y + 5 z)
8. 9 x^2 + 49 y^2 + 16 z^2 + 42 x y + 56 y z + 24 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }9 x^2 + 49 y^2 + 16 z^2 + 42 x y + 56 y z + 24 z x
(3 x)^2 + (7 y)^2 + (4 z)^2 + 2 * 3 x * 7 y + 2 * 7 y * 4 z + 2 * 4 z * 3 x
(3 x + 7 y + 4 z)^2
(3 x + 7 y + 4 z) (3 x + 7 y + 4 z)
9. 4 x^2 + 4 y^2 + 4 z^2 - 8 x y + 8 y z - 8 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }4 x^2 + 4 y^2 + 4 z^2 - 8 x y + 8 y z - 8 z x
(2 x)^2 + (2 y)^2 + (2 z)^2 + 2 * 2 x * (-2 y) + 2 * (-2 y) * (-2 z) + 2 * (-2 z) * 2 x
(2 x - 2 y - 2 z)^2
(2 x - 2 y - 2 z) (2 x - 2 y - 2 z)
10. 9 x^2 + 9 y^2 + 9 z^2 - 18 x y + 18 y z - 18 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }9 x^2 + 9 y^2 + 9 z^2 - 18 x y + 18 y z - 18 z x
(-3 x)^2 + (3 y)^2 + (3 z)^2 + 2 * (-3 x) * 3 y + 2 * 3 y * 3 z + 2 * 3 z * (-3 x)
(-3 x + 3 y + 3 z)^2
(-3 x + 3 y + 3 z) (-3 x + 3 y + 3 z)
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| maths-ncert-solution |
11. x^2 + 4 y^2 + 9 z^2 - 4 x y - 12 y z + 6 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }x^2 + 4 y^2 + 9 z^2 - 4 x y - 12 y z + 6 z x
(x)^2 + (2 y)^2 + (3 z)^2 + 2 * x * (-2 y) + 2 * (-2 y) * 3 z + 2 * 3 z * x
(x - 2 y + 3 z)^2
(x - 2 y + 3 z) (x - 2 y + 3 z)
12. 4 x^2 + 9 y^2 + z^2 + 12 x y + 6 y z + 4 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }4 x^2 + 9 y^2 + z^2 + 12 x y + 6 y z + 4 z x
(-2 x)^2 + (-3 y)^2 + (-z)^2 + 2 * (-2 x) * (-3 y) + 2 * (-3 y) * (-z) + 2 * (-z) * (-2 x)
(-2 x - 3 y - z)^2
(-2 x - 3 y - z) (-2 x - 3 y - z)
13. 9 x^2 + 4 y^2 + z^2 - 12 x y + 4 y z - 6 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }9 x^2 + 4 y^2 + z^2 - 12 x y + 4 y z - 6 z x
(3 x)^2 + (-2 y)^2 + (-z)^2 + 2 * 3 x * (-2 y) + 2 * (-2 y) * (-z) + 2 * (-z) * 3 x
(3 x - 2 y - z)^2
(3 x - 2 y - z) (3 x - 2 y - z)
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| maths-ncert-solution |
14. x^2 + 4 y^2 + 49 z^2 + 4 x y - 28 y z - 14 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }x^2 + 4 y^2 + 49 z^2 + 4 x y - 28 y z - 14 z x
(x)^2 + (2 y)^2 + (-7 z)^2 + 2 * x * 2 y + 2 * 2 y * (-7 z) + 2 * (-7 z) * x
(x + 2 y + 7 z)^2
(x + 2 y + 7 z) (x + 2 y + 7 z)
15. 4 x^2 + 4 y^2 + 25 z^2 + 8 x y - 20 y z - 20 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }4 x^2 + 4 y^2 + 25 z^2 + 8 x y - 20 y z - 20 z x
(-2 x)^2 + (-2 y)^2 + (5 z)^2 + 2 * (-2 x) * (-2 y) + 2 * (-2 y) * 5 z + 2 * 5 z * (-2 x)
(-2 x - 2 y + 5 z)^2
(-2 x - 2 y + 5 z) (-2 x - 2 y + 5 z)
16. 9 x^2 + 49 y^2 + 16 z^2 - 42 x y - 56 y z + 24 z x
Sol. By using identity
{ (x + y + z)^2 = x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x }
9 x^2 + 49 y^2 + 16 z^2 - 42 x y - 56 y z + 24 z x
(-3 x)^2 + (7 y)^2 + (-4 z)^2 + 2 * (-3 x) * 7 y + 2 * 7 y * (-4 z) + 2 * (-4 z) * (-3 x)
(-3 x + 2 y + z)^2
(-3 x + 2 y + z) (-3 x + 2 y + z)
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