Quadratic, Logarithm and Power Formulas

Formulas for the quadratic equation, logarithms and exponents with electrical examples: real and complex roots, decibels, powers and roots.

The formulas

The discriminant D tells you the kind of roots: positive gives two real roots, zero gives one repeated root, and negative gives a complex-conjugate pair. When b is much larger than a and c, the plain formula loses digits in the small root to cancellation; find the large root first and the small one from x₁ × x₂ = c / a. A logarithm needs a positive number and a base that is positive and not 1. An even root of a negative number is not real, but an odd root is (the cube root of −8 is −2).

Worked example: real roots

Solve x² − 5x + 6 = 0.

1. D = (−5)² − 4 × 1 × 6 = 1
2. x = (5 ± √1) / 2, so x₁ = 3 and x₂ = 2
3. Check: 3 + 2 = 5 = −b / a and 3 × 2 = 6 = c / a

Worked example: complex roots

Solve x² + 2x + 5 = 0, as in the characteristic equation of a damped series circuit.

1. D = 2² − 4 × 1 × 5 = -16, which is negative
2. Real part = −b / 2a = -1 and imaginary part = √16 / 2 = 2
3. The roots are -1 + j2 and -1 − j2

Worked example: decibels

1. A power ratio of 100 is 10 × log₁₀(100) = 20 dB
2. A voltage ratio of 2 is 20 × log₁₀(2) = 6.021 dB
3. Powers and roots: 2¹⁰ = 1024, and the cube root of 27 is 3

Frequently asked questions

What does a negative discriminant mean?

The equation has no real roots. The two roots are a complex-conjugate pair, and the parabola does not cross the x axis. In a circuit this is the underdamped case, which oscillates.

Why is the logarithm of zero or a negative number not allowed?

No power of a positive base gives zero or a negative number, so there is no real logarithm. The logarithm of a number approaching zero falls without limit.

Why is decibels 10 log for power and 20 log for voltage?

Power goes with the square of voltage or current. Since log(x²) = 2 log x, a voltage ratio uses 20 times the logarithm to give the same decibel value as the matching power ratio.

Can I take the square root of a negative number?

Not as a real number. The result is imaginary, so use the complex number calculator. An odd root of a negative number, such as the cube root of −8, is real.

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