Given that a + b = 2c,
determine which of the following expressions is true.

Question

Given that a + b = 2c,
 determine which of the following expressions is true.

Accepted Answer

Given a + b = 2c.

We need to check which expression is true.

Check option 3: a² + 2bc = b² + 2ac

Rewrite the equation:

Left side: a² + 2bc
Right side: b² + 2ac

Bring all terms to one side:

(a² + 2bc) - (b² + 2ac) = 0
=> a² - b² + 2bc - 2ac = 0
=> (a² - b²) + 2c(b - a) = 0

Factor a² - b²:

(a - b)(a + b) + 2c(b - a) = 0

Note that (b - a) = -(a - b), so rewrite 2c(b - a) as -2c(a - b):

(a - b)(a + b) - 2c(a - b) = 0

Factor (a - b):

(a - b)(a + b - 2c) = 0

Given a + b = 2c, so (a + b - 2c) = 0.

Therefore, (a - b)*0 = 0, which is true.

Hence, option 3 is true.

Checking other options similarly will show they do not satisfy the equation for all values.

Therefore, the correct expression is option 3: a² + 2bc = b² + 2ac.

Answer

Answer

Answer