- Thread starter acemi123
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Note that x

I like solving quadratic inequalities by considering the graph. There are also more algebraic ways.

What would the graph of y = x

When will this graph be below the x-axis?

Be systematic.solve inequation with 2 unknown! What is the way to solve this type of inequations?

x^2+4ax<0

a-) 0<x<4a for every a Real number

b-) no solution if a>0

c-) -4a<x<0 if a>0

d-) -4a<x<0 if a<0

Thanks.

Solve the corresponding equality first.

\(\displaystyle x^2 + 4ax = 0 \implies x(x + 4a) = 0 \implies x = 0 \text { or } x = - \ 4a.\)

So you know that

\(\displaystyle x = 0 \implies x^2 + 4ax \not < 0 \text {, and } x = -\ 4a \not < 0.\)

So you have four cases to to consider, namely:

(1) x > 0 and x > - 4a, (2) x > 0 and x < - 4a, (3) x < 0 and x > - 4a, and (4) x < 0 and x < - 4a.

Can you figure it out now?

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x^2- 4ax= x(x- 4a)$. In order that the product of two number be negative, one must be positive and the other negative so

(i) x< 0, x- 4a> 0

(ii) x> 0, x- 4a< 0.

In case (i) x> 4a so that x< 0 can only be true if 4a is negative. If a is negative then 4a< x< 0.

In case (ii) x< 4a so that x> 0 can only be true if 4a is positive. If a is positive then 0< x< 4a.