State the following statement is True or False
The parabola $f (x) = a x^{2} + b x + c$ does not touch or intersect with the x axis if $b^{2} < \sqrt{3} a c$

True

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False

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Solution

The correct option is A True
$b^{2} < \sqrt{3} a c < 4 a c$
$∴ b^{2} - 4 a c < 0$
So, the equation $a x^{2} + b c + c = 0$ has no real solution.
Which means that $f (x) = a x^{2} + b c + c$ does not touch the x-axis.
So, the given statement is true.

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State the following statement is True or FalseThe parabola f(x)=ax2+bx+c does not touch or intersect with the x axis if b2<√3ac

The correct option is A Trueb2<√3ac<4ac∴b2−4ac<0So, the equation ax2+bc+c=0 has no real solution.Which means that f(x)=ax2+bc+c does not touch the x-axis.So, the given statement is true.

State the following statement is True or False
The parabola $f (x) = a x^{2} + b x + c$ does not touch or intersect with the x axis if $b^{2} < \sqrt{3} a c$

The correct option is A True
$b^{2} < \sqrt{3} a c < 4 a c$
$∴ b^{2} - 4 a c < 0$
So, the equation $a x^{2} + b c + c = 0$ has no real solution.
Which means that $f (x) = a x^{2} + b c + c$ does not touch the x-axis.
So, the given statement is true.