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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
Table of content
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 =.
