Solve each equation in Exercises 83–108 by the method of your choice. $x^{2} - 6 x + 13 = 0$

검증된 단계별 안내

Identify the type of equation: This is a quadratic equation in the form $x^2 - 6x + 13 = 0$, where $a = 1$, $b = -6$, and $c = 13$.

Calculate the discriminant using the formula $\Delta = b^2 - 4ac$. Substitute the values to get $\Delta = (-6)^2 - 4(1)(13)$.

Evaluate the discriminant to determine the nature of the roots: if $\Delta > 0$, there are two real roots; if $\Delta = 0$, one real root; if $\Delta < 0$, two complex roots.

Since the discriminant is less than zero, use the quadratic formula to find the complex roots: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$.

Substitute $a$, $b$, and $\Delta$ into the quadratic formula and simplify to express the solutions in terms of real and imaginary parts.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.

Discriminant and Nature of Roots

The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If it is positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots.

Solving Quadratic Equations Using the Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation. It is especially useful when factoring is difficult or impossible, and it accounts for all types of roots based on the discriminant.

Solve each equation in Exercises 83–108 by the method of your choice. x2−6x+13=0x^2 - 6x + 13 = 0