Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 64a

2장, 문제 64a

Solve each equation in Exercises 47–64 by completing the square. 3x² - 5x - 10 = 0

검증된 단계별 안내

Rewrite the equation in standard quadratic form, isolating the constant term on one side:

3x^2 - 5x = 10

Divide through by the coefficient of

x^2

(which is 3) to make the coefficient of

x^2

equal to 1:

x^2 - \(\frac{5}{3}\)x = \(\frac{10}{3}\)

To complete the square, take half the coefficient of

x

(which is

-\(\frac{5}{3}\)

), square it, and add it to both sides of the equation. Half of

-\(\frac{5}{3}\)

-\(\frac{5}{6}\)

, and squaring it gives

\(\frac{25}{36}\)

. Add

\(\frac{25}{36}\)

to both sides:

x^2 - \(\frac{5}{3}\)x + \(\frac{25}{36}\) = \(\frac{10}{3}\) + \(\frac{25}{36}\)

Simplify the left-hand side into a perfect square trinomial and combine the fractions on the right-hand side. The left-hand side becomes

(x - \(\frac{5}{6}\))^2

, and the right-hand side simplifies to a single fraction.

Take the square root of both sides of the equation, remembering to include both the positive and negative roots. Then solve for

x

by isolating it:

x = \(\frac{5}{6}\) \(\pm\) \(\sqrt{\text{(simplified fraction)}\)}

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주요 개념

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

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves rearranging the equation and adding a specific value to both sides to create a square of a binomial. This technique simplifies the process of finding the roots of the equation and is particularly useful when the quadratic formula is not preferred.

Quadratic Equations

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 these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the standard form and properties of quadratic equations is essential for solving them effectively.

Discriminant

The discriminant is a component of the quadratic formula, given by the expression b^2 - 4ac. It determines the nature of the roots of a 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. Analyzing the discriminant helps in understanding the solutions without necessarily solving the equation.

Solve each equation in Exercises 47–64 by completing the square. 3x2 - 5x - 10 = 0