Question

Solve the quadratic equation. $$7 z^{2}-46 z=21$$

Short answer

The solutions to the quadratic equation are \(z=7\) and \(z=-3/7\).

Step-by-step solution

Rewrite the Equation in Standard Form

Move all terms to one side of the equation to achieve standard form: \(7 z^{2} - 46 z - 21 = 0\). Now we can define the coefficients a = 7, b = -46, and c = -21 in the quadratic formula.

Substitute a, b and c into the Quadratic Formula

Substitute a, b, and c into the quadratic formula. \(z = [-(-46) ± sqrt{(-46)^{2}-4*7*(-21)}]/ (2*7)\). Simplify under the square root to get \( z = [46 ± sqrt{(2116) + (588)}] / 14\).

Simplify and Evaluate z

Simplify further to find \( z = [46 ± sqrt{(2704)}]/14 = [46 ± 52] / 14\). This gives us two solutions: \( z = 98/14 = 7\) and \( z = -6/14 = -3/7\).