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.

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 = (36)/(5) and a+c = 15. Substitute these values into the equation 2ac = b(a+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 = (36)/(5) and a+c = 15. Substitute these values into the equation 2ac = b(a+c).

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.

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.

If the equation $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ has equal roots, where $a + c = 15$ and $b = \frac{36}{5}$ , then $a^2 + c^2$ is equal to

Answer: $a^2 + c^2 = 144.4$

Step-by-step solution

Step 1: Apply condition for equal roots

For a quadratic equation of the form $Ax^2 + Bx + C = 0$ 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 $B^2 - 4AC = 0$ . 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 $b = \frac{36}{5}$ and $a+c = 15$ . Substitute these values into the equation $2ac = b(a+c)$ .

Step 6: Calculate $a^2 + c^2$

We need to find $a^2 + c^2$ . We know the value of $a+c$ and we just calculated $2ac$ . We can use the algebraic identity $(a+c)^2 = a^2 + c^2 + 2ac$ to find $a^2 + c^2$ . Rearranging the identity, we get $a^2 + c^2 = (a+c)^2 - 2ac$ .

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If the equation a(b−c)x2+b(c−a)x+c(a−b)=0a(b - c)x^2 + b(c - a)x + c(a - b) = 0a(b−c)x2+b(c−a)x+c(a−b)=0 has equal roots, where a+c=15a + c = 15a+c=15 and b=365b = \frac{36}{5}b=536​, then a2+c2a^2 + c^2a2+c2 is equal to