Question

find all solutions by factoring.
2t² - 9t = 5

Explanation:

Step1: Rewrite the equation in standard form

Subtract 5 from both sides to get \(2t^{2}-9t - 5=0\).

Step2: Factor the quadratic expression

We need two numbers that multiply to \(2\times(- 5)=- 10\) and add up to - 9. The numbers are - 10 and 1.
Rewrite the middle term: \(2t^{2}-10t + t-5 = 0\).
Group the terms: \((2t^{2}-10t)+(t - 5)=0\).
Factor out the common factors: \(2t(t - 5)+1(t - 5)=0\).
Then we have \((2t + 1)(t - 5)=0\).

Step3: Solve for t

Set each factor equal to zero:

• \(2t+1 = 0\) gives \(2t=-1\), so \(t=-\frac{1}{2}\).
• \(t - 5=0\) gives \(t = 5\).

Answer:

\(t = 5\) or \(t=-\frac{1}{2}\)