If the equation a(b−c)x2+b(c−a)x+c(a−b)=0 has equal roots, where a+c=15 and b=536, then a2+c2 is equal to
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Step-by-Step Solution
Step 1: Apply condition for equal roots
For a quadratic equation of the form Ax2+Bx+C=0 to have equal roots, its discriminant D must be equal to zero. In this problem, A=a(b−c), B=b(c−a), and C=c(a−b).
Step 2: Substitute coefficients into discriminant equation
Substitute the given coefficients A, B, and C into the discriminant formula B2−4AC=0. This gives us an equation relating a, b, and c.
Step 3: Simplify the equation
Expand and simplify the equation obtained in the previous step. This involves squaring the term b(c-a) and multiplying the terms a(b-c) and c(a-b).
Step 4: Use the property for equal roots
A special property of quadratic equations states that if the sum of the coefficients A+B+C=0, then x=1 is a root. If the roots are equal, then x=1 must be the repeated root. Let's check if A+B+C=0 for the given equation. If A+B+C=0, then a(b−c)+b(c−a)+c(a−b)=ab−ac+bc−ab+ac−bc=0. Since the sum of the coefficients is zero, the roots are equal, and x=1 is the repeated root. This implies that b(c−a)2−4ac(b−c)(a−b)=0 is satisfied.
Step 5: Substitute given values and solve for a and c
Since the roots are equal, and the sum of coefficients is zero, we know that x=1 is the repeated root. This means that the quadratic equation can be written as A(x−1)2=0. Comparing the constant term, C=A. So, c(a−b)=a(b−c). Expanding this, we get ac−bc=ab−ac. Rearranging, we have 2ac=ab+bc=b(a+c). We are given b=536 and a+c=15. Substitute these values into the equation 2ac=b(a+c).
Step 6: Calculate a2+c2
We need to find a2+c2. We know the value of a+c and we just calculated 2ac. We can use the algebraic identity (a+c)2=a2+c2+2ac to find a2+c2. Rearranging the identity, we get a2+c2=(a+c)2−2ac.