Questions: Solve for (t). [ d=-16 t^2+15 t ] What is the answer? ( t=frac-15 pm sqrt225-64 d-32 )

Solve for (t).
[ d=-16 t^2+15 t ]

What is the answer?
( t=frac-15 pm sqrt225-64 d-32 )
Transcript text: Solve for $t$. \[ d=-16 t^{2}+15 t \] What is the answer? $t=\frac{-15 \pm \sqrt{225-64 d}}{-32}$
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Solution

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Solution Steps

To solve for t t in the equation d=16t2+15t d = -16t^2 + 15t , we recognize it as a quadratic equation in the form at2+bt+c=0 at^2 + bt + c = 0 . We can use the quadratic formula t=b±b24ac2a t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} to find the values of t t . Here, a=16 a = -16 , b=15 b = 15 , and c=d c = -d .

Step 1: Identify the Quadratic Equation

The given equation is d=16t2+15t d = -16t^2 + 15t . This is a quadratic equation in the standard form at2+bt+c=0 at^2 + bt + c = 0 , where a=16 a = -16 , b=15 b = 15 , and c=d c = -d .

Step 2: Apply the Quadratic Formula

To solve for t t , we use the quadratic formula: t=b±b24ac2a t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Substituting the values of a a , b b , and c c , we have: t=15±1524(16)(d)2(16) t = \frac{-15 \pm \sqrt{15^2 - 4(-16)(-d)}}{2(-16)}

Step 3: Calculate the Discriminant

The discriminant is given by: b24ac=1524(16)(d)=22564d b^2 - 4ac = 15^2 - 4(-16)(-d) = 225 - 64d For d=0 d = 0 , the discriminant simplifies to: 225 225

Step 4: Solve for t t

Substitute the discriminant back into the quadratic formula: t=15±22532 t = \frac{-15 \pm \sqrt{225}}{-32} Calculate the two possible values for t t : t1=15+1532=0 t_1 = \frac{-15 + 15}{-32} = 0 t2=151532=0.9375 t_2 = \frac{-15 - 15}{-32} = 0.9375

Final Answer

t=15±22564d32\boxed{t = \frac{-15 \pm \sqrt{225 - 64d}}{-32}}

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