Question

Find the number of integral values of 'x' for which 'y' is not negative, given that y = -2x2 + 3x + 7.
{x' is a real number)
ОО
O2
04
O 5

Answer 1

Answer:

The number of integral values of x for which y is not negative is 4.

Option 3) 4

Step-by-step-explanation:

We have given that,

For the equation - 2x² + 3x + 7 = y, y is not negative.

And x is a real number.

We have to find the number of integral value of x.

y is not negative.

$\displaystyle{\therefore\:\sf\:y\:\geq\:0}$

$\displaystyle{\implies\sf\:-\:2x^2\:+\:3x\:+\:7\:\geq\:0}$

Comparing with $\displaystyle{\sf\:ax^2\:+\:bx\:+\:c\:\geq\:0}$, we get,

• a = - 2
• b = 3
• c = 7

Now,

$\displaystyle{\sf\:b^2\:-\:4ac\:=\:3^2\:-\:4\:(\:-\:2\:\times\:7\:)}$

$\displaystyle{\implies\sf\:b^2\:-\:4ac\:=\:9\:-\:4\:(\:-\:14\:)}$

$\displaystyle{\implies\sf\:b^2\:-\:4ac\:=\:9\:+\:56}$

$\displaystyle{\implies\:\boxed{\blue{\sf\:b^2\:-\:4ac\:=\:65\:}}}$

Now, by quadratic formula,

$\displaystyle{\boxed{\pink{\sf\:x\:\geq\:\dfrac{\:-\:b\:\pm\:\sqrt{b^2\:-\:4ac}}{2a}\:}}}$

$\displaystyle{\implies\sf\:x\:\geq\:\dfrac{-\:3\:\pm\:\sqrt{65}}{2\:\times\:(\:-\:2\:)}}$

$\displaystyle{\implies\sf\:x\:\geq\:\dfrac{-\:3\:\pm\:\sqrt{65}}{-\:4}}$

$\displaystyle{\implies\sf\:x\:\geq\:\dfrac{-\:(\:-\:3\:\pm\:\sqrt{65}\:)}{4}}$

$\displaystyle{\implies\sf\:x\:\geq\:\dfrac{3\:\mp\:\sqrt{65}}{4}}$

$\displaystyle{\implies\sf\:x\:\geq\:\dfrac{3\:-\:\sqrt{65}}{4}\:\quad\:OR\:\quad\:x\:\geq\:\dfrac{3\:+\:\sqrt{65}}{4}}$

$\displaystyle{\implies\sf\:\dfrac{3\:-\:\sqrt{65}}{4}\:\leq\:x\:\geq\:\dfrac{3\:+\:\sqrt{65}}{4}}$

$\displaystyle{\implies\sf\:\dfrac{3\:-\:8.062}{4}\:\leq\:x\:\geq\:\dfrac{3\:+\:8.062}{4}}$

$\displaystyle{\implies\sf\:\dfrac{-\:5.062}{4}\:\leq\:x\:\geq\:\dfrac{11.062}{4}}$

$\displaystyle{\implies\sf\:-\:1.2655\:\leq\:x\:\geq\:2.7655}$

$\displaystyle{\implies\sf\:x\:\in\:[\:-\:1.2655\:,\:2.7655\:]}$

Taking the integral part, we get,

$\displaystyle{\implies\sf\:x\:\in\:[\:-\:1\:,\:2\:]}$

$\displaystyle{\implies\:\underline{\boxed{\red{\sf\:x\:=\:\{\:-\:1\:,\:0\:,\:1\:,\:2\:\}\:}}}}$

∴ The number of integral values of x for which y is not negative is 4.