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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 116
Chapter 2, Problem 116

Find all values of x satisfying the given conditions. y = 5x2 + 3x and y = 2

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Start by setting the two equations equal to each other since both represent y. This gives: 5x^2 + 3x = 2.
Rearrange the equation to set it equal to 0, which is the standard form for a quadratic equation: 5x^2 + 3x - 2 = 0.
Identify the coefficients of the quadratic equation: a = 5, b = 3, and c = -2. These will be used in the quadratic formula.
Apply the quadratic formula: x = \(\frac{-b \pm \sqrt{b^2 - 4ac}\)}{2a}. Substitute the values of a, b, and c into the formula.
Simplify the discriminant b^2 - 4ac and then compute the two possible solutions for x by evaluating the formula with both the plus and minus options for the square root.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c. In this case, the function y = 5x^2 + 3x represents a parabola that opens upwards, where 'a' is positive. Understanding the properties of quadratic functions, such as their vertex, axis of symmetry, and roots, is essential for solving equations involving them.
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Finding Intersections

Finding the intersection of two functions involves determining the values of x where the functions are equal. In this problem, we set the quadratic function equal to the constant function y = 2 to find the x-values that satisfy both equations. This process often requires rearranging the equation into standard form and applying methods such as factoring, completing the square, or using the quadratic formula.
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The Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0. It provides the solutions for x based on the coefficients a, b, and c. In this context, once the equation is rearranged to standard form, applying the quadratic formula will yield the x-values where the two functions intersect.
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Related Practice
Textbook Question

Find all values of x satisfying the given conditions. y1 = x - 1, y2 = x + 4 and y1y2 = 14

Textbook Question

Find all values of x satisfying the given conditions. y = 2x2 - 3x and y = 2

Textbook Question

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

Textbook Question

Find all values of x satisfying the given conditions. y1 = 2x2 + 5x - 4, y2 = - x2 + 15x - 10, and y1 - y2 = 0

Textbook Question

Find all values of x satisfying the given conditions. y1 = 2x/(x + 2), y2 = 3/(x + 4), and y1 + y2 = 1

Textbook Question

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 = - (x + 1)2 + 4

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