Problem 99 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Write a quadratic equation in standard form whose Solutions are \(-3\) and 5

Step-by-Step Solution

Answer

The quadratic equation in standard form whose solutions are -3 and 5 is \(x^2 - 2x - 15 = 0\).

Step 1: Writing the equation by given roots

The roots of the quadratic equation are given as -3 and 5. Using the relationship between the roots and the standard form of a quadratic equation, the equation becomes \(a(x-(-3))(x-5) = a(x+3)(x-5) = 0\)

Step 2: Expanding the equation

The equation \(a(x+3)(x-5) = 0\) can be expanded to give the quadratic equation in standard form. Distributing the terms we obtain \(ax^2 - 5ax + 3ax - 15a = 0\). Further simplifying by combining like terms we get the equation \(ax^2 - 2ax - 15a = 0\)

Step 3: Writing the Quadratic Equation in Standard Form

The standard form of the quadratic equation is \(ax^2 + bx + c = 0\). Comparing this with our derived equation \(ax^2 - 2ax - 15a = 0\), we get \(b = -2a\) and \(c = -15a\). To get rid of the coefficients of \(b\) and \(c\), we can choose \(a = 1\) (considering \(a \neq 0\) as for a quadratic equation \(a\) should not be zero). Substituting \(a =1\) into the equation gives the equation in standard form as \(x^2 - 2x - 15 = 0\).

Key Concepts

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation is essential for solving and analyzing these second-degree polynomials. The standard form is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a \) is not equal to zero. It's because if \( a \) were zero, the equation would no longer be quadratic but a linear one.

In this form, we can easily identify the properties of the quadratic equation, such as its axis of symmetry, vertex, and direction of opening (upward if \( a > 0 \) or downward if \( a < 0 \)). It is crucial to express quadratic equations in this form to apply various techniques for finding solutions, such as factoring, completing the square, or using the quadratic formula.

Factoring Quadratic Equations

Factoring is a method to solve quadratic equations that involves breaking down the quadratic into a product of simpler expressions that can be set to zero and solved individually. For a quadratic equation in standard form, \( ax^2 + bx + c = 0 \), the factoring process aims to find two binomials, \((x-p)\) and \((x-q)\), such that when multiplied they return the original quadratic equation and the solutions are the roots \( p \) and \( q \).

To factor a quadratic equation, one must look for two numbers that multiply to \( ac \) (the product of the coefficient of \( x^2 \) and the constant term) and add to \( b \) (the coefficient of \( x \)). However, when the leading coefficient \( a \) is not 1, the process becomes more complex and might involve techniques such as grouping or the use of the quadratic formula to determine the roots. Factoring is a preferred method when the quadratic expression can easily be decomposed into factors.

Quadratic Equation Roots

The roots of a quadratic equation represent the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). In other words, roots are the solutions to the equation, where the graph of the equation intersects with the x-axis.

There are a few ways to find the roots of a quadratic equation. One such way is by factoring, as mentioned earlier. Another common method is using the quadratic formula: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \), where the discriminant \( b^2 - 4ac \) determines the nature of the roots (real and distinct, real and repeated, or complex). If the discriminant is positive, there are two real distinct roots; if it's zero, there is one repeated real root; if negative, the roots are complex.

Roots are also closely tied to the graphical representation of the quadratic; if one were to sketch the graph, the roots would correspond to the points where the parabola crosses the x-axis. Understanding and determining roots are fundamental to various applications of quadratic equations, such as optimization problems and parabolic motion in physics.