Question

Solve the equation. Tell which solution method you used. \(t^{2}-16 t+65=0\)

Short answer

The solutions to the equation \(t^{2}-16 t+65=0\) are \(8 + i\) and \(8 - i\). The quadratic formula was used to solve for 't'.

Step-by-step solution

Identify the coefficients

In the quadratic equation, the coefficients are as follows: a = 1, b = -16, and c = 65.

Apply the Quadratic Formula

The quadratic formula is \(t = [-b \pm sqrt(b^{2} - 4ac)]/2a \). Substitute the coefficients into this formula and solve for 't'.

Simplify the Expression Inside the Square Root

Calculate the value inside the square root, which is \(b^{2} - 4ac = (-16)^{2} - 4*1*65 = 256-260 = -4\).

Deal with the Negative Inside the Square Root

The value within the square root is negative. We can work with this by identifying it as \(sqrt(-1) * sqrt(4)\), where \(sqrt(-1)\) is represented as imaginary unit 'i' and \(sqrt(4)\) equals 2.

Calculate the Roots

Substitute the calculated square root into the quadratic formula. So, \(t = [16 \pm 2i]/2 = 8 \pm i\)