Mathematics: Solve for t. use the quadratic formula. d=−16t^2+12t

Solve for t. use the quadratic formula. d=−16t^2+12t

beverlyamya

15.03.2022

$t\,=\,\frac{-3+\sqrt{9+4d}}{-8}\:\:and\:\:\frac{3+\sqrt{9+4d}}{8}$

Step-by-step explanation:

Given: d = -16t² + 12t

To find: t using quadratic formula

If we have quadratic equation in form ax² + bx + c = 0

then, by quadratic formula we have

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

Rewrite the given equation,

-16t² + 12t - d = 0

from this equation we have,

a = -16 , b = 12 , c = d

now using quadratic formula we get,

$t\,=\,\frac{-12\pm\sqrt{12^2-4\times(-16)\times d}}{2\times(-16)}$

$t\,=\,\frac{-12\pm\sqrt{144+64d}}{-32}$

$t\,=\,\frac{-12\pm\sqrt{16(9+4d)}}{-32}$

$t\,=\,\frac{-12\pm4\sqrt{9+4d}}{-32}$

$t\,=\,\frac{4(-3\pm\sqrt{9+4d})}{-32}$

$t\,=\,\frac{-3\pm\sqrt{9+4d}}{-8}$

$t\,=\,\frac{-3+\sqrt{9+4d}}{-8}\:\:,\:\:\frac{-3-\sqrt{9+4d}}{-8}$

$t\,=\,\frac{-3+\sqrt{9+4d}}{-8}\:\:and\:\:\frac{-(3+\sqrt{9+4d})}{-8}$

$t\,=\,\frac{-3+\sqrt{9+4d}}{-8}\:\:and\:\:\frac{3+\sqrt{9+4d}}{8}$

Therefore, $t\,=\,\frac{-3+\sqrt{9+4d}}{-8}\:\:and\:\:\frac{3+\sqrt{9+4d}}{8}$

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