Question

Q. 4 Solve the following quadratic equations by using completing square method. (i) a^(2)-5a-10=0 (ii) 12r^(2)-17r-5=0

Answer 1

(i)
`\(a^2 - 5a - 10 = 0\)` can be solved using the completing the square method, yielding
`\(a = (5 + √(65))/(2)\) or \(a = (5 - √(65))/(2)\).`

(ii) For
`\(12r^2 - 17r - 5 = 0\),` completing the square gives
`\(r = (17 + √(289 + 240))/(24)\) or \(r = (17 - √(289 + 240))/(24)\)`, which simplifies to
`\(r = (17 + √(529))/(24)\) or \(r = (17 - √(529))/(24)\), resulting in \(r = (17 + 23)/(24)\) or \(r = (17 - 23)/(24)\).`

Step-by-step explanation:

To solve the quadratic equation
`\(a^2 - 5a - 10 = 0\)`, we use the completing the square method. First, move the constant term to the other side, yielding
`\(a^2 - 5a = 10\)`. Next, add
`\(\left((5)/(2)\right)^2\)` to both sides to create a perfect square trinomial, resulting in
`\((a - (5)/(2))^2 = (65)/(4)\).` Taking the square root of both sides gives
`\(a - (5)/(2) = \pm (√(65))/(2)\)`, leading to the solutions
`\(a = (5 + √(65))/(2)\) or \(a = (5 - √(65))/(2)\).`

For the equation
`\(12r^2 - 17r - 5 = 0\),` applying the completing the square method involves isolating the quadratic and linear terms, resulting in
`\(12r^2 - 17r = 5\).` Completing the square, we add
`\(\left((17)/(2)\right)^2\) to both sides, leading to \((2r - (17)/(2))^2 = (529)/(4)\). Taking the square root gives \(2r - (17)/(2) = \pm (23)/(2)\),` which simplifies to
`\(r = (17 + 23)/(24)\) or \(r = (17 - 23)/(24)\).`

In both cases, this method provides the solutions with precise calculation, ensuring accuracy in solving quadratic equations.