# If roots of the equation (a – b)x2 + (b – c)x + (c – a) = 0 are equal, prove that 2a = b + c.

By BYJU'S Exam Prep

Updated on: September 25th, 2023

It is given that the quadratic equation is (a – b)x2 + (b – c)x + (c – a) = 0. The quadratic equation’s discriminant is equal to zero because the roots are equal. Now we have to compare the given quadratic equation with the standard form which is Ax2 + Bx + C = 0.

Then we get, A = (a – b), B = (b – c), C = (c – a)

Therefore, discriminant

D = B2 – 4AC = 0

On rearranging we get:

B2 = 4AC

Substituting the values we get:

(b – c)2 = 4(a – b) (c – a)

Expanding the above equation using the formula:

b2 + c2 -2bc = 4 (ac – a2 – bc + ab)

b2 + c2 -2bc = 4ac – 4a2 – 4bc + 4ab

b2 + c2 + 4a2 + 2bc – 4ac – 4ab = 0

b2 + c2 + (-2a)2 + 2bc + 2(-2a)c + 2(-2a)b = 0

In simplification we get the:

(b + c – 2a)2 = 0

b + c – 2a = 0

2a = b + c

### Roots of the Equation

• A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign (=).
• Two expressions are combined in an equation using an equal symbol (=).
• The left-hand side and
ight-hand side of the equation are the two expressions on either side of the equals sign.
• Typically, we consider an equation’s right side to be zero.
• Since we can balance this by deducting the right-side expression from both sides’ expressions, this won’t reduce the generality.

Summary:

## If roots of the equation (a – b)x2 + (b – c)x + (c – a) = 0 are equal, prove that 2a = b + c.

If the roots of the equation (a – b)x2 + (b – c)x + (c – a) = 0 are equal, it is proved that 2a = b + c. In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

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