Find the quadratic equation whose roots are α+5β and β+5α where α and β are roots of the quadratic equation 2x2+7x−3=0

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Ourst1977 · Accepted Answer

Step 1  Given quadratic equation is 2x2+7x−3=0  α and β are the roots of the given quadratic equation.  We know that sum of roots of a quadratic equation =−ba  So, α+β=−72  Product of roots =ca  ⇒αβ=−32  Step 2  Now, we have to find quadratic equation whose roots are α+5β and β+5α  The quadratic equation whose roots are given then equation of quadratic is given as:  x2−(sum of roots)x+(product of roots)=0  Therefore, the required quadratic equation is:  x2−(α+5β+β+5α)x+(α+5β×β+5α)=0  x2−(α(α+5)+β(β+5)αβ)x+(α(β+5)+5(β+5)αβ)=0  x2−(α2+5α+β2+5βαβ)x+(αβ+5α+5β+25αβ)=0  x2−((α+β)2−αβ+5(α+β)αβ)x+(αβ+5(α+β)+25αβ)=0  Step 3  Now, putting the value of (α+β) and αβ in above equation, we get  x2−((−72)2−2×−32+5×−72−32)x+(−−32+5×−72+25−32)=0  x2−(494+3−352−32)x+(−32−352+25−32)=0

Find the quadratic equation whose roots are \frac{\alpha+5}{\beta} and \frac{\beta+5}{\alpha}

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