Question

ASAP!! Please help me. I will not accept nonsense answers, but will mark as BRAINLIEST if you answer is correctly with solutions.

Answer 1

Answer:

The function has two real roots and crosses the x-axis in two places.

The solutions of the given function are

x = (-0.4495, 4.4495)

Step-by-step explanation:

The given quadratic equation is

$G(x) = -x^2 + 4x + 2$

A quadratic equation has always 2 solutions (roots) but the nature of solutions might be different depending upon the equation.

Recall that the general form of a quadratic equation is given by

$a^2 + bx + c$

Comparing the general form with the given quadratic equation, we get

$a = -1 \\\\b = 4\\\\c = 2$

The nature of the solutions can be found using

If $b^2- 4ac = 0$ then we get two real and equal solutions

If $b^2- 4ac > 0$ then we get two real and different solutions

If $b^2- 4ac < 0$ then we get two imaginary solutions

For the given case,

$b^2- 4ac \\\\(4)^2- 4(-1)(2) \\\\16 - (-8) \\\\16 + 8 \\\\24$

Since 24 > 0

we got two real and different solutions which means that the function crosses the x-axis at two different places.

Therefore, the correct option is the last one.

The function has two real roots and crosses the x-axis in two places.

The solutions (roots) of the equation may be found by using the quadratic formula

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{-(4)\pm\sqrt{(4)^2-4(-1)(2)}}{2(-1)} \\\\x=\frac{-4\pm\sqrt{(16 - (-8)}}{-2} \\\\x=\frac{-4\pm\sqrt{(24}}{-2} \\\\x=\frac{-4\pm 4.899}{-2} \\\\x=\frac{-4 + 4.899}{-2} \: and \: x=\frac{-4 - 4.899}{-2}\\\\x= -0.4495 \: and \: x = 4.4495$

Therefore, the solutions of the given function are

x = (-0.4495, 4.4495)

A graph of the given function is also attached where you can see that the function crosses the x-axis at these two points.