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Chapter 1: Problem 2

Solve each equation in Exercises \(1-14\) by factoring. $$ x^{2}-13 x+36=0 $$

Short Answer

Expert verified
The solutions to the equation \(x^2 - 13x + 36 = 0\) by factoring are \(x = 9\) and \(x = 4\).

Step by step solution

01

Set Up the Problem

We have the quadratic equation \(x^2 - 13x + 36 = 0\). This is in the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -13\), and \(c = 36\).
02

Factoring the Quadratic

The quadratic expression \((x^2 - 13x + 36)\) can be factored by finding two numbers that multiply to \(36\) and add up to \(-13\). These numbers are \(-9\) and \(-4\). Therefore, the factored form of the expression is \((x - 9)(x - 4) = 0\).
03

Solving for x

A product of factors equals zero if and only if at least one of the factors equals zero. Therefore, we can set each factor equal to zero and solve for \(x\): \n1) \(x - 9 = 0\) gives \(x = 9\). \n2) \(x - 4 = 0\) gives \(x = 4\).

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Key Concepts

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

Quadratic Equations
Quadratic equations form a cornerstone in mathematics. These equations are typically in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). When compared to linear equations, quadratics involve the square of the variable, \(x\).
This introduces a parabolic graph shape. Quadratics can have zero, one, or two real solutions because the graph can intersect the x-axis at these possible points.
Understanding a quadratic equation means knowing the role each component plays:
  • \('a'\) - Determines the parabola's direction (upwards if positive, downwards if negative)
  • \('b'\) - Affects the position of the vertex horizontally
  • \('c'\) - Influences where the parabola crosses the y-axis
The ultimate goal is to identify the values of \(x\) that satisfy the equation, which are also known as the roots of the equation.
Factoring
Factoring is a pivotal technique in algebra used to simplify expressions and solve equations. To factor a quadratic, like \(x^2 - 13x + 36\), you seek two numbers that both multiply to \(c\) (here, 36) and sum to \(b\) (here, -13).
The process often includes:
  • Identifying the product-sum pair. In our example, \(-9\) and \(-4\) multiply to \(36\) and add to \(-13\).
  • Expressing the quadratic as a product of binomials. The equation becomes \((x-9)(x-4)\).
Factoring transforms a quadratic into a form that is more manageable to work with. Not every quadratic can be factored easily, especially if the roots are not whole numbers, but this approach remains one of the first steps attempted in solving quadratics.
Solving Equations
Solving equations involves finding the value(s) of the variable that makes the equation true. For factored quadratic equations, such as \((x-9)(x-4)=0\), the key principle is that if a product equals zero, then at least one factor must be zero.
This principle allows us to set each binomial factor equal to zero to solve for \(x\):
  • For \(x-9=0\), solving gives \(x=9\).
  • For \(x-4=0\), solving gives \(x=4\).
Thus, \(x=9\) and \(x=4\) are the solutions. These solutions represent the x-values where the parabola represented by \(x^2 - 13x + 36 = 0\) intersects the x-axis.
Factoring simplifies the process of finding these critical points in comparison to other methods like completing the square or using the quadratic formula.

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Most popular questions from this chapter

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. A rectangular park is 4 miles long and 2 miles wide. How long is a pedestrian route that runs diagonally across the park?

A rectangular parking lot has a length that is 3 yards greater than the width. The area of the parking lot is 180 square yards. Find the length and the width.

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$D=R T \quad \text{for} \quad R$$

Explain what it means to solve a formula for a variable.

An isosceles right triangle has legs that are the same length and acute angles each measuring \(45^{\circ} .\) (GRAPH NOT COPY) a. Write an expression in terms of \(a\) that represents the length of the hypotenuse. b. Use your result from part (a) to write a sentence that describes the length of the hypotenuse of an isosceles right triangle in terms of the length of a leg.
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