Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 86

Chapter 4, Problem 86

A quadratic equation ƒ(x) = 0 has a solution x = 2. Its graph has vertex (5, 3). What is the other solution of the equation?

Verified step by step guidance

Recall that the quadratic function can be written in vertex form as $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Here, the vertex is given as $(5, 3)$, so the function can be expressed as $f(x) = a(x - 5)^2 + 3$.

Since $x = 2$ is a solution to the equation $f(x) = 0$, substitute $x = 2$ and $f(x) = 0$ into the vertex form to find the value of $a$: $0 = a(2 - 5)^2 + 3$.

Simplify the equation to solve for $a$: calculate $(2 - 5)^2$ and then isolate $a$ on one side of the equation.

Once $a$ is found, write the quadratic function explicitly as $f(x) = a(x - 5)^2 + 3$.

To find the other solution, set $f(x) = 0$ and solve the equation $a(x - 5)^2 + 3 = 0$ for $x$. This will give two solutions, one of which is $x = 2$, and the other will be the other solution you are asked to find.

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Key Concepts

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

Quadratic Equation and Its Roots

A quadratic equation is a second-degree polynomial of the form ax² + bx + c = 0. It has two solutions or roots, which can be real or complex. These roots correspond to the x-values where the graph of the quadratic function intersects the x-axis.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Knowing the vertex helps in understanding the shape and position of the parabola, and it can be used to find the quadratic equation when combined with other points.

Symmetry of a Parabola

A parabola is symmetric about a vertical line called the axis of symmetry, which passes through the vertex. If one root is known, the other root can be found by reflecting it across the axis of symmetry, using the vertex's x-coordinate as the midpoint between the roots.

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Related Practice

Textbook Question

Solve each inequality. Give the solution set in interval notation. 4/(x+6)>2/(x-1)

Textbook Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $ƒ (x) = x^{5} + 3 x^{4} - x^{3} + 2 x + 3$

Textbook Question

Graph each rational function. ƒ(x)=(20+6x-2x²)/(8+6x-2x²)

Textbook Question

Graph each rational function. ƒ(x)=(18+6x-4x²)/(4+6x+2x²)

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Textbook Question

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. ƒ(x)=2x³-5x²-x+1; [1.4, 2]

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Textbook Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $ƒ (x) = 6 x^{4} + 2 x^{3} + 9 x^{2} + x + 5$