Question

If x – y = c, z – x = b, y – z = a, a^2 + b^2 + c^2 =25, then the value of x^2 + y^2 + z^2– xy – yz – zx is equal to

Answer 1

Step-by-step explanation:

x - y = c ------- 1

z - x = b --------2

y - z = a --------3

a²+b²+c² = 25 ---------------- 4

x²+y²+z²-xy-yz-zx = ?

putting values of eqn 1,2 and 3 in eqn 4

(y-z)² + (z-x)² + (x-y)² = 25

*a²-2ab+b² = (a-b)² applying this formula in above eqn

{y²-2yz+z²}+{z²-2zx+x²}+{ x²-2xy+y²} = 25

y²-2yz+z²+z²-2zx+x²+x²-2xy+y² = 25

2x²+2y²+2z²-2xy-2yz-2zx = 25

taking 2 common from above eqn

2 (x²+y²+z²-xy-yz-zx) = 25

x²+y²+z²-xy-yz-zx = 25*2

x²+y²+z²-xy-yz-zx = 50. ANSWER

Answer 2

25/2

Given:

x - y = c

z - x = b

y - z = a

a² + b² + c² = 25

To find:

The value of x² + y² + z² - xy - yz - xz

Answer in 3 Steps

1. Square all terms by Using Algebra Formula.

2. Substitute the given value.

3. Simplify and Get answer.

Solution:

x - y = c

[Squaring both side]

(x - y)² = c²

x² + y² - 2xy = c²

Similarly, By Squaring (z - x) and (y - z) I get,

z² + x² - 2xz = b² and y² + z² - 2yz = a²

Now, Adding all terms,

x² + y² - 2xy + z² + x² - 2xz +y² + z² - 2yz

= a²+ b² +c²

2x² + 2y² + 2z² - 2xy - 2xz - 2yz = 25

Or, 2(x² + y² + z² - xy - xz - yz )= 25

Or, x² + y² + z² - xy - xz - yz = 25/2

Therefore, The required answer is 25/2

Some Important Algebra Formula

${(x + y)}^{2}={x}^{2}+{y}^{2}+2xy\\ \\{(x - y)}^{2}={x}^{2}+{y}^{2}-2xy\\ \\{(x+y)}^{2}= (x - y) {}^{2}+4xy\\ \\{(x-y)}^{2}=(x+y){}^{2}-4xy\\ \\ (x + y)^{2}+(x-y)^{2}=2( {x}^{2}+{y}^{2} )\\ \\(x+y)^{2}- (x-y) {}^{2}=4xy\\\\ (x+y)(x-y)= x^2 - y^2$