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jarptica [38.1K]
2 years ago
14

If z^1/2 = x^1/2 + y^1/2 , prove that (x + y - z)^2 = 4xy.HELPP MEEEEE ​

Mathematics
2 answers:
lawyer [7]2 years ago
5 0
The answer is 6 cap use photo math
S_A_V [24]2 years ago
4 0

Answer:

Make use of the fact that (a + b)^{2} = a^{2} + 2\, a\, b + b^{2}.

Step-by-step explanation:

Start from the given: z^{1/2} = x^{1/2} + y^{1/2}.

In other words: \sqrt{z} = \sqrt{x} + \sqrt{y}.

Square both sides of the equation:

(\sqrt{z})^{2} = (\sqrt{x} + \sqrt{y})^{2}.

Equality would still hold because:

\begin{aligned}& a = b \\ \implies \; & a\cdot a = a\cdot b \\ \implies \; & a \cdot a = b \cdot b \\ \implies \; & a^{2} = b^{2}\end{aligned}.

Simplify:

z = (\sqrt{x})^{2} + 2\, (\sqrt{x}) \cdot (\sqrt{y}) + (\sqrt{y})^{2}.

z = x + 2\sqrt{x y} + y.

Subtract z from both sides of this equation:

0 = x + 2 \sqrt{x y}  + y - z.

Subtract 2 \sqrt{x y} from both sides of this equation:

-2 \sqrt{x y} = x + y - z.

Again, square both sides of this equation:

(-2 \sqrt{x y})^{2} = (x + y - z)^{2}.

(-2)^{2}\, (\sqrt{x y})^{2} = (x + y - z)^{2}.

(x + y - z)^{2} = 4\, x \, y (by the symmetric property of equalities.)

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