Problem 39 Solve. SEE EXAMPLE $3 .$ (OBJE... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 39

Solve. SEE EXAMPLE $3 .$ (OBJECTIVE 1 ) $$x^{2}-x-20=0$$

Short Answer

Expert verified

The solutions to the equation $x^{2} - x - 20 = 0$ are $x = 5$ and $x = -4$.

Step by step solution

Identify the coefficients

In the equation $x^{2} - x - 20 = 0$, the coefficient of $x^{2}$ (a) is 1, the coefficient of $x$ (b) is -1, and the constant term (c) is -20.

Factorize the quadratic equation

If the quadratic equation can be factorized, it would look like this: $(x - p)(x - q) = 0$. In this case, we need to find two numbers that multiply to -20 (which is $ac$) and add up to -1 (which is $b$). The numbers suitable for this are -5 and 4, since $-5 * 4 = -20$ and $-5 + 4 = -1$. Therefore, the factorized form of the equation is $(x - 5)(x + 4) = 0$.

Solve each factor for the variable

Now, we equate each factor to zero and solve for $x$, which gives us our roots. So it would be $x - 5 = 0$ (which gives $x = 5$) and $x + 4 = 0$ (which gives $x = -4$).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorization

Factorization is a crucial method used for solving quadratic equations. It simplifies equations into a product of simpler forms, making it easier to find solutions. The primary goal of factorization in a quadratic equation like $x^2 - x - 20 = 0$ is to express it as a product, such as $(x - p)(x - q) = 0$. This simplifies the equation substantially.

First, identify the necessary coefficients: the coefficient of $x^2$ (generally called $a$), the coefficient of $x$ ($b$), and the constant term ($c$). For the given equation, $a = 1$, $b = -1$, and $c = -20$.
The factorization process involves finding two numbers that multiply to produce the product $ac = (-20)$ and add up to $b = -1$. In our example, these numbers are -5 and 4.
Finally, rewrite the quadratic equation as $(x - 5)(x + 4) = 0$.

This factorization breaks down the original quadratic equation, paving the way for easier solutions.

Solving Quadratic Equations

Once the quadratic equation is factorized, the next step is solving it. The factorization has reduced the complexity of the problem, and now we can find solutions by solving each factor separately.

Since $(x - 5)(x + 4) = 0$ is equivalent to saying either $x - 5 = 0$ or $x + 4 = 0$, you simply solve for $x$ in each case.
From $x - 5 = 0$, adding 5 to both sides yields $x = 5$.
From $x + 4 = 0$, subtracting 4 from both sides gives you $x = -4$.

Solving these simpler equations results in the roots of the original quadratic equation. This step-by-step approach is effective and straightforward.

Roots of Equations

The roots of a quadratic equation are the values of $x$ that make the equation equal to zero. For the equation $x^2 - x - 20 = 0$, we have already found these roots through solving the factors.

The roots of the factorized equation $(x - 5)(x + 4) = 0$ are $x = 5$ and $x = -4$.
These roots correspond to the x-intercepts of the graph of the quadratic equation. When plotted, the quadratic function will intersect the x-axis at these points.
Understanding these roots helps in analyzing the behavior of quadratic functions graphically and numerically.

Having recognized the roots of the equation, one appreciates the practical value of solving quadratic equations by factorization and similar techniques.

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91影视

Solve. SEE EXAMPLE \(3 .\) (OBJECTIVE 1 ) $$x^{2}-x-20=0$$

Short Answer

Step by step solution

Identify the coefficients

Factorize the quadratic equation

Solve each factor for the variable

Key Concepts

Factorization

Solving Quadratic Equations

Roots of Equations

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