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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 1 - Equations and Inequalities문제 113
2장, 문제 113

In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = x2 - 2x + 2

검증된 단계별 안내
1
To find the x-intercepts of the graph of the equation \( y = x^2 - 2x + 2 \), set \( y = 0 \). This is because the x-intercepts occur where the graph crosses the x-axis, and at these points, the value of \( y \) is zero.
The equation becomes \( 0 = x^2 - 2x + 2 \). This is a quadratic equation, so we will solve it using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a \), \( b \), and \( c \) are the coefficients from the standard form of the quadratic equation \( ax^2 + bx + c = 0 \).
Identify the coefficients: \( a = 1 \), \( b = -2 \), and \( c = 2 \). Substitute these values into the quadratic formula: \( x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(2)}}{2(1)} \).
Simplify the discriminant (the expression under the square root): \( (-2)^2 - 4(1)(2) = 4 - 8 = -4 \). Since the discriminant is negative, the equation has no real solutions, meaning the graph does not cross the x-axis. Instead, the solutions are complex numbers.
Conclude that the graph of \( y = x^2 - 2x + 2 \) has no x-intercepts. This means the parabola does not intersect the x-axis, and its vertex lies above the x-axis because the parabola opens upwards (as \( a > 0 \)).

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주요 개념

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

X-Intercept

The x-intercept of a graph is the point where the graph intersects the x-axis. This occurs when the value of y is zero. To find the x-intercept(s) of an equation, you set y equal to zero and solve for x. In the context of the given equation, this means solving the quadratic equation x^2 - 2x + 2 = 0.
추천 영상:
04:08
Graphing Intercepts

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 is not zero. The solutions to a quadratic equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the nature of the roots (real or complex) is essential for determining the x-intercepts.
추천 영상:
05:35
Introduction to Quadratic Equations

Graphing Quadratics

Graphing a quadratic function involves plotting a parabola, which can open upwards or downwards depending on the sign of the leading coefficient (a). The vertex of the parabola represents the maximum or minimum point, and the x-intercepts indicate where the graph crosses the x-axis. Analyzing the graph helps in visualizing the solutions and understanding the behavior of the function.
추천 영상:
06:36
Solving Quadratic Equations Using The Quadratic Formula
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