Question

★ CONTENT QUALITY SUPPORT REQUIRED ★

★ STANDARD QUESTION 57 ★

The answer with highest efficiency will get marked THE Brainliest , And will be offered an special invitation card

Question is -

For what Values of a is the inequality ax² + 2ax + 0.5 > 0 valid throughout the entire number axis


This question needs a perfect answer ; and it doesn't support lazy ones ! Take care

Answer 1

$given \: a {x}^{2} + 2ax + 0.5 \\ differenciate \: above \: statement \\ 2ax + 2a = 0 \\ x = - 1 \\ the \: given \: eqn \: has \: a \: turning \: point \\ and \: touches \: xaxis \: at \: x = - 1 \\ a {x}^{2} + 2ax + 0.5 > 0 \: if \: \\ coefficient \: of \: {x}^{2} > 0 \\ \: if \: a > 0 \: the \: curve \: willnot \: enter \: \\ into \: negative \: y \: axis \\ so \: the \: eqn \: is \: > 0 \: for \: all \: a > 0$
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Answer 2

Answer:

0 < a < 1/2

Step-by-step explanation:

since the quadratic expression is positive for all real x, so

D<0 & a>0

=>4a²-4a(0.5) < 0

=>4a²-2a < 0

=>2a(2a-1) < 0

using wavy-curve method,

.'. 0<a<1/2

which also satisfies a>0