Solve each quadratic equation using the quadratic formula. See Example 7. x² - 4x + 3 = 0

Verified step by step guidance

1. Identify the coefficients from the quadratic equation \(x^2 - 4x + 3 = 0\). Here, \(a = 1\), \(b = -4\), and \(c = 3\).
2. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
3. Calculate the discriminant \(\Delta = b^2 - 4ac\). Substitute the values: \(\Delta = (-4)^2 - 4 \times 1 \times 3\).
4. Evaluate the square root of the discriminant: \(\sqrt{\Delta}\), which will be used in the formula.
5. Substitute \(b\), \(a\), and \(\sqrt{\Delta}\) into the quadratic formula to find the two possible values of \(x\): \(x = \frac{-(-4) \pm \sqrt{\Delta}}{2 \times 1}\).

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It represents a parabola when graphed and has up to two real or complex solutions.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses the coefficients a, b, and c to find the roots, including real and complex solutions.

Discriminant

The discriminant, given by Δ = b² - 4ac, determines the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real roots; if Δ = 0, one real root; and if Δ < 0, two complex conjugate roots.