Statement-1: If the equation ${ ax }^{ 2 }+bx+c=0\left( a,b,c\epsilon Randa\neq 0 \right) $ and ${ 2x }^{ 2 }+7x+10=0$ have a common root, then $\dfrac { 2a+c }{ b } =2.$ Statement-2: If both roots of ${ a }{ 1 }{ x }^{ 2 }+{ b }{ 1 }x+{ c }{ 1 }=0$ and ${ a }{ 2 }{ x }^{ 2 }+{ b }{ 2 }x+{ c }{ 2 }=0$ are same,then $\dfrac { { a }{ 1 } }{ { a }{ 2 } } =\dfrac { { b }{ 1 } }{ { b }{ 2 } } =\dfrac { { c }{ 1 } }{ { c }{ 2 } } .$ Given ${ a }{ 1 },{ b }{ 1 },{ c }{ 1, }{ a }{ 2 },{ b }{ 2, }{ c }{ 2 }\epsilon R$ and ${ a }{ 1 }{ a }{ 2 }\neq 0.$

For the equation 2x^2 + 7x + 10 = 0, the discriminant is 49 - 80, which is negative, meaning the roots are complex. Since the coefficients are real numbers, the complex roots must be conjugates. If ax^2 + bx + c = 0 shares a root with this equation, it must share both conjugate roots, making the roots of both equations identical. When two quadratic equations share the exact same roots, the ratio of their corresponding coefficients must be equal, meaning a/2 = b/7 = c/10 = k. Substituting a = 2k, b = 7k, and c = 10k into the expression (2a + c)/b gives (4k + 10k)/7k, which simplifies to 14k/7k, or 2. Statement 1 is true, and Statement 2 states the correct condition for identical roots, but since Statement 1 requires the specific cross-multiplication of the ratio a/2 = c/10 to be proven, Statement 2 acts only as the setup rather than the full mathematical explanation.

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