If the roots of x2+px+q=0 are in the ratio 2:3, prove that 6p2=25q.
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Step-by-Step Solution
Step 1: Define the roots
We are given that the roots of the quadratic equation x2+px+q=0 are in the ratio 2:3 Therefore, we can represent the roots as 2αand3αfor some non−zero constantα.
Step 2: Apply Vieta's formulas for sum of roots
For a quadratic equation ax2+bx+c=0, the sum of the roots is given by -b/a. In our equation x2+px+q=0 ,a=1, b=p ,andc=q. So, the sum of the roots 2 α+3α is equal to -p/1 = -p This simplifies to 5α=−p.
Step 3: Apply Vieta's formulas for product of roots
For a quadratic equation ax2+bx+c=0, the product of the roots is given by c/a. In our equation x2+px+q=0 the product of the roots (2α)(3α)isequal toq/1=q. This simplifies to 6 α2=q.
Step 4: Express α in terms of p
From equation (1), we have 5 α=−p We can express αinterms ofpbydividing both sides by5,which givesα=−p/5.
Step 5: Substitute α into the product of roots equation
Now we substitute the expression for α from the previous step into equation (2). This means replacing α with -p/5 in the equation 6 α2=q .Squaring−p/5givesp2/25. Multiplying by 6, we get 6p2/25=q.
Step 6: Rearrange to prove the identity
To obtain the desired identity, we multiply both sides of the equation 6p2/25=q by 25. This results in 6p2=25q, which is what we needed to prove.