Solution

=Hints for difficult problems = $$p^3+q^3$$ Pre requisites: Interpretation of the Problem: Concepts: Algorithm: Consider the equation $$2a^2-4a+1=0$$ Here a=2,b=-4 & c=1 If p & q are the roots of the quadratic equation then $$p+q={\frac{-b}{a}}={\frac{-(-4)}{2}=2}$$ $$pq={\frac{c}{a}}={\frac{1}{2}}$$ Therefore, $$p^3+q^3=(p+q)^3-3pq(p+q)$$ =$$(2)^3-3[{\frac{1}{2}}](2)$$ =8-3 =5 to back to concept page [| quadratic equation problems] 2.The altitude of a triangle is 6cm greter than its base. If its area is 108cmsq .Find its base. solution Statement: Solving problem based on quadratic equations. solution 216=x2+6x x2 +6x -216=0 Substitution: x 2 +18x-12x -216=0 Simplification: x(x+18)-12(x+18)=0 (x+18)( x-12)=0 (x+18)=0 (x-12)=0 x=-18, x=12.
 * 1) If P & q are the roots of the equation $$2a^2-4a+1=0$$ find the value of
 * 1) Standard form of quadratic equation
 * 2) Formula to find the sum & product of quadratic equation
 * 3) Knowledge of using appropriate identity
 * 1) Compare the equation with standard form and identify the values of a,b,c
 * 2) To find the sum formformof the roots of the quadratic equation using the formula
 * 3) To find the product of the roots of the equation
 * 4) Using the identity & rewriting $$p^3+q^3$$ as $$(p+q)^3-3pq(p+q)$$
 * 5) Substitute the values of m+n & mn in $$(p+q)^3-3pq(p+q)$$
 * 6) Simplification
 * 1) Formula to find the sum and product of the roots of the quadratic equation
 * 2) Identity $$(a+b)^3=a^3+b^3+3ab(a+b)$$
 * Interpretation of the problem: * Converting data in to eqn. *Knowledge about area of a triangle. *knowledge of the formula of area of triangle. *Methods of finding the roots of the eqn. *Methods of finding the roots of the
 * Different approches to solve the problem: *Factorisation
 * Using formula
 * using graph
 * Concept used:Forming the eqn. 216=x(x+6)
 * 1) Base=12cm,  Altitude=x+6

=12+6=18cm. Prior Knowledge - back to concept page [| quadratic equation problems]
 * Methods of solving the Eqn
 * Factorisation
 * Using Formula
 * Using Graph