Equations - Tips

# For an equation, if all the even powers of x have same sign coefficients and all the odd powers of x have the opposite sign coefficients, then it has no negative roots

# For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x).

# For a cubic equation ax3+bx2+cx+d=o

· Sum of the roots = - b/a

· Sum of the product of the roots taken two at a time = c/a

· Product of the roots = -d/a


For a bi-quadratic equation ax4+bx3+cx2+dx+e = 0

· Sum of the roots = - b/a

· Sum of the product of the roots taken three at a time = c/a

· Sum of the product of the roots taken two at a time = -d/a

· Product of the roots = e/a


1. Consider the two equations

a1x+b1y=c1

a2x+b2y=c2



Then,

- If a1/a2 = b1/b2 = c1/c2, then we have infinite solutions for these equations.

- If a1/a2 = b1/b2 <> c1/c2, then we have no solution.

- If a1/a2 <> b1/b2, then we have a unique solution.



2. Roots of x2 + x + 1=0 are 1, w, w2 where 1 + w + w2=0 and w3=1



3. |a| + |b| = |a + b| if a*b>=0

else, |a| + |b| >= |a + b|



4. The equation ax2+bx+c=0 will have max. value when a<0 and min. value when a>0. The max. or min. value is given by (4ac-b2)/4a and will occur at x = -b/2a



5. If for two numbers x + y=k (a constant), then their PRODUCT is MAXIMUM if x=y (=k/2). The maximum product is then (k2)/4.

If for two numbers x*y=k (a constant), then their SUM is MINIMUM if
x=y (=root(k)). The minimum sum is then 2*root (k).



6. Product of any two numbers = Product of their HCF and LCM. Hence product of two numbers = LCM of the numbers if they are prime to each other.

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