Solve each equation in Exercises 65–74 using the quadratic formula. x² - 6x + 10 = 0

Verified step by step guidance

1. Identify the coefficients of the quadratic equation in standard form ax^2 + bx + c = 0. Here, a = 1, b = -6, and c = 10.
2. Recall the quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. Substitute the values of a, b, and c into the formula.
3. Simplify the discriminant, which is the part under the square root: $b^2-4ac$. Compute ${-}^6$ (b squared) and subtract $4ac$.
4. Determine whether the discriminant is positive, zero, or negative. If it is negative, the solutions will involve imaginary numbers. If it is zero, there is one real solution. If it is positive, there are two distinct real solutions.
5. Simplify the entire expression for x by calculating the numerator and dividing by the denominator. If the discriminant is negative, express the solutions in terms of complex numbers using $i$, where $i=\sqrt{-1}$.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula.

Quadratic Formula

The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation. It is expressed as x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients from the quadratic equation. This formula provides the roots of the equation, which can be real or complex depending on the value of the discriminant (b² - 4ac).

Discriminant

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots of the quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. Understanding the discriminant helps in predicting the type of solutions without solving the equation.