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Chapter 6: Problem 44

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(x(x-3)=18\)

Short Answer

Expert verified
The solutions to the equation are \(x = 6\) and \(x = -3\).

Step by step solution

01

Expand the Equation

Expand the equation \(x(x-3)=18\), which simplifies to \(x^2 - 3x = 18\)
02

Rearrange the Equation

Set the equation to zero, \(x^2 - 3x - 18 = 0\). This will enable you to factor the equation.
03

Factor the Equation

Begin factoring the quadratic equation by identifying factors of 18 that add up to -3. So, \(x^2 - 6x + 3x -18 = 0\). Regroup the equation to factor by grouping, \(x(x - 6) + 3(x - 6) = 0\). Ultimately, this simplifies down to \((x - 6)(x + 3) = 0\).
04

Solve for x

Set each factor equal to zero in order to solve for x. \(x - 6 = 0 => x = 6\) and \(x + 3 = 0 => x = -3\)

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Key Concepts

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

Factoring
In mathematics, factoring is a powerful technique used to simplify expressions or solve equations, particularly quadratic equations. To factor a quadratic equation means to express it as a product of simpler expressions, often referred to as factors. In the quadratic equation, the general form is expressed as \(ax^2 + bx + c\). The goal is to rewrite it as \((mx + n)(px + q)\).

Here鈥檚 how it works:
  • Identify two numbers that will multiply to give you the constant (the last term, "c") and add to give you the middle coefficient ("b").
  • Rewrite the quadratic equation as two sets of binomial expressions.
  • Use the zero-product property, which states that if the product is zero, at least one of the multipliers must be zero, to find the solutions.
Factoring is invaluable because it breaks down complex expressions into simpler parts, making it easier to analyze and solve. This was the approach used to solve the equation in the original exercise.
Roots of Quadratic
The roots of a quadratic equation are the values of \(x\) that make the equation equal to zero. These values are also referred to as zeros or x-intercepts when graphing. Finding the roots is equivalent to solving the equation. In our specific example, after factoring \((x - 6)(x + 3) = 0\), the roots can be quickly identified by setting each factor equal to zero.

Here鈥檚 how it unfolds:
  • From \((x - 6) = 0\), solve for \(x = 6\).
  • From \((x + 3) = 0\), solve for \(x = -3\).
These roots are points where the quadratic crosses the x-axis. Understanding the roots helps in not only solving equations but also in modeling real-world scenarios where they indicate significant values.
Polynomial Function
A polynomial function represents a sum of multiple algebraic terms. In the context of quadratics, these functions specifically involve equations of degree two. The standard form of a quadratic polynomial function is \(f(x) = ax^2 + bx + c\).

Characteristics of quadratic polynomial functions include:
  • They always shape into a parabola when graphed.
  • The leading coefficient \(a\) determines the parabola's direction: if \(a > 0\), it opens upwards, if \(a < 0\), it opens downwards.
  • The vertex of the parabola gives the maximum or minimum point of the function.
In solving polynomial functions like our quadratic equation, understanding these properties helps in predicting the graph鈥檚 shape and behavior, facilitating graphical or substitution checks to confirm solutions.
Graphing Techniques
Graphing techniques serve as a visual representation of algebraic equations like quadratic ones. For students, seeing a graph helps solidify the relationships among the solutions and the equation itself. When graphing a quadratic equation, the parabola's key features鈥攙ertex, axis of symmetry, and x-intercepts (roots)鈥攂ecome apparent.

To graph a quadratic function, one would:
  • Start by identifying the vertex using \((-b/2a, f(-b/2a))\).
  • Plot the roots, or x-intercepts, where the equation equals zero.
  • Draw the axis of symmetry that runs vertically through the vertex.
  • Sketch the parabola using the identified features.
A graphing utility can quickly sketch this parabola, displaying where it intercepts the x-axis, which are precisely the roots obtained by factoring. This acts as a secondary verification for the solutions derived algebraically.

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Most popular questions from this chapter

Use the \(x\)-intercepts for the graph in \(a[-10,10,1]\) by \([-13,10,1]\) viewing rectangle to solve the quadratic equation. Check by substitution. Use the graph of \(y=x^{2}+x-6\) to solve \(x^{2}+x-6=0\).

Exercises \(137-139\) will help you prepare for the material covered in the next section. In each exercise, factor completely. $$2 x^{2}-20 x+50$$

What is a quadratic equation?

Why is \(\frac{6 x+12}{7 x-28}\) undefined for \(x=4 ?\)

Solve equation and check your solutions. \((x-4)\left(x^{2}+5 x+6\right)=0\)
See all solutions

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