Solve using the quadratic formula 2x² - 2x = 1.

Solution:

Given quadratic equation : 2x² - 2x = 1

⇒2x² - 2x - 1 = 0

If the quadratic equation is of the form ax² + bx + c = 0 then roots(zeros) of the quadratic equation are given by x = [-b ± √(b² - 4ac)] / 2a

Here, b² - 4ac = Δ where Δis called discriminant that determines the nature of roots of the given quadratic equation.

(i) If Δ > 0, then roots are real and distinct

(ii) If Δ= 0, then roots are real and repeated

(iii) If Δ < 0, then roots are imaginary or complex.

Compare 2x² - 2x - 1 = 0 with ax² + bx + c = 0 

∴ a = 2, b = -2 and c = -1

∴ The roots are x = [-b ± √(b² - 4ac)] / 2a = [-(-2) ± √((-2)² - 4(2)(-1))] / 2(2)

x = [2 ± √12] / 4 

x = [2 ± 2√3] / 4 

x = [1 ± √3] / 2

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Solve using the quadratic formula 2x² - 2x = 1.

Summary:

The roots of the quadratic formula 2x² - 2x = 1 are x = [1 ± √3]/ 2.