Question

use the quadratic formula to solve. express your answer in simplest form. 8t² ± 26t + 15 = 0

Explanation:

Step1: Identify coefficients

For quadratic equation \(at^2 + bt + c = 0\), here \(a = 8\), \(b = -26\) (assuming the symbol is minus, maybe a typo, as \(+\) and \(\pm\) together is odd; let's take \(8t^2 - 26t + 15 = 0\)), \(c = 15\).

Step2: Quadratic formula

Quadratic formula: \(t=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Substitute \(a = 8\), \(b = -26\), \(c = 15\):
\(t=\frac{-(-26)\pm\sqrt{(-26)^2 - 4\times8\times15}}{2\times8}\)

Step3: Calculate discriminant

\((-26)^2 - 4\times8\times15 = 676 - 480 = 196\)

Step4: Simplify square root

\(\sqrt{196}=14\)

Step5: Solve for t

\(t=\frac{26\pm14}{16}\)
Case 1: \(t=\frac{26 + 14}{16}=\frac{40}{16}=\frac{5}{2}\)
Case 2: \(t=\frac{26 - 14}{16}=\frac{12}{16}=\frac{3}{4}\)

If the middle term was \(+26t\), then \(b = 26\), discriminant same:
\(t=\frac{-26\pm14}{16}\)
Case 1: \(t=\frac{-26 + 14}{16}=\frac{-12}{16}=-\frac{3}{4}\)
Case 2: \(t=\frac{-26 - 14}{16}=\frac{-40}{16}=-\frac{5}{2}\)
Assuming the correct equation is \(8t^2 - 26t + 15 = 0\) (more likely, as positive constant and middle term sign might be minus due to typo).

Answer:

\(t = \frac{5}{2}\) or \(t = \frac{3}{4}\) (if \(b=-26\)) or \(t = -\frac{3}{4}\) or \(t = -\frac{5}{2}\) (if \(b = 26\))