Question

sandoval, alexzander
unit 4 formative geometry sy 25 - 26
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place the number that correctly completes the sentence.
a rocket blasted through the air and followed a path defined by the function ( g ( x ) = - 2 x ^ { 2 } - 6 x + 80 ), where ( g ( x ) ) is the height of the rocket after ( x ) seconds.
it takes the rocket seconds to hit the ground.

Explanation:

Step1: Set the height function equal to zero

When the rocket hits the ground, \(g(x) = 0\). So we have the equation \(-2x^{2}-6x + 80=0\). Multiply through by \(-1\) to get \(2x^{2}+6x - 80 = 0\). Divide by \(2\) to simplify: \(x^{2}+3x - 40=0\).

Step2: Factor the quadratic equation

We need to find two numbers that multiply to \(-40\) and add up to \(3\). The numbers are \(8\) and \(-5\). So the factored form is \((x + 8)(x - 5)=0\).

Step3: Solve for \(x\)

Set each factor equal to zero: \(x+8 = 0\) gives \(x=-8\), and \(x - 5=0\) gives \(x = 5\). Since time \(x\) cannot be negative, we discard \(x=-8\).

Answer:

\(5\)