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Solve (8t-2)/(5) = (2)/(t)

July 24, 2026Mathematics

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Answer

t = \frac{5}{4}, -1

4 steps

To solve the equation $\frac{8t - 2}{5} = \frac{2}{t}$ , we will cross-multiply and then solve the resulting quadratic equation.

Cross-multiply the terms.
$t(8t - 2) = 5(2)$ $8t^2 - 2t = 10$
Rearrange the equation into the standard quadratic form $at^2 + bt + c = 0$ .
$8t^2 - 2t - 10 = 0$
Divide the entire equation by 2 to simplify the coefficients:
$4t^2 - t - 5 = 0$
Solve the quadratic equation using the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
For $4t^2 - t - 5 = 0$ , we have $a = 4$ , $b = -1$ , and $c = -5$ .
$t = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(-5)}}{2(4)}$ $t = \frac{1 \pm \sqrt{1 - (-80)}}{8}$ $t = \frac{1 \pm \sqrt{1 + 80}}{8}$ $t = \frac{1 \pm \sqrt{81}}{8}$ $t = \frac{1 \pm 9}{8}$
Calculate the two possible values for $t$ .
$t_1 = \frac{1 + 9}{8} = \frac{10}{8} = \frac{5}{4}$ $t_2 = \frac{1 - 9}{8} = \frac{-8}{8} = -1$
The solutions for $t$ are $\frac{5}{4}$ and $-1$ .
$t = \frac{5}{4}, -1$