Problem 35 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$5 x^{2}=18-x$$

Step-by-Step Solution

Answer

The solution to the quadratic equation are \(x = 9/5\) and \(x = -2\).

Step 1: Arrange into Standard Form

In order to factor the equation, first rearrange the equation into the standard form \(Ax^2 + Bx + C = 0\). In this case, the equation becomes \(5x^2 + x - 18 = 0\) after adding \(x\) and subtracting \(18\) from both sides.

Step 2: Factor the Quadratic Equation

You need to think of two numbers that multiply to \(-90\) (i.e., \(5*-18\)) and add up to \(1\). The numbers are \(10\) and \(-9\). Thus, the quadratic equation can be factored as \((5x - 9)(x + 2) = 0\). After performing the necessary computations and manipulations, the quadratic equation rearranges to: \(5x^2 - 9x + 2x - 18 = 0\)

Step 3: Solve for x

Now, set each factor equal to zero and solve for \(x\) respectively. That is, for \(5x - 9 = 0\), which gives \(x = 9/5\) and for \(x + 2=0\), which gives \(x = -2\).

Key Concepts

Standard Form of a Quadratic

The standard form of a quadratic equation is an essential starting point for solving, graphing, and understanding its properties. It is expressed as \(Ax^2 + Bx + C = 0\), where \(A\), \(B\), and \(C\) are constants, and \(A\) is not equal to zero. This format allows us to compare all quadratic equations uniformly.

When faced with an equation like \(5x^2 = 18 - x\), the first step is to rearrange terms to achieve the standard form. This process involves moving all terms to one side of the equation to set it equal to zero, resulting in \(5x^2 + x - 18 = 0\). This step is a crucial foundation for the subsequent processes of factoring, solving, and graphing the quadratic equation.

Solving Quadratic Equations

Solving quadratic equations means finding the values of \(x\) that satisfy the equation. There are several methods to solve quadratics, but one of the most fundamental ways is factoring. To factor, we look for two binomials that multiply to give the original quadratic equation. In our example, \(5x^2 + x - 18= 0\), the suitable binomials are \((5x - 9)(x + 2) = 0\), because the product of the outer and inner terms of the binomials gives the middle term \(Bx\) of the standard form, and their constant terms multiply to \(AC\).

Once factored, we solve for \(x\) by setting each binomial equal to zero – known as the zero-product property. The solutions are the \(x\)-intercepts of the graph of the quadratic function. In this case, we find two solutions: \(x = 9/5\) and \(x = -2\). These solutions are critical, as they are the points where the graph of the quadratic equation will intersect the \(x\)-axis.

Graphing Quadratic Functions

Graphing quadratic functions gives a visual representation of the solutions to a quadratic equation and reveals its parabolic shape. The graph of a standard quadratic equation \(Ax^2 + Bx + C\) is a curve called a parabola, which opens upwards if \(A\) is positive and downwards if \(A\) is negative. To graph the quadratic function, you plot points for various \(x\)-values and connect them to form the parabola.

The vertex of the parabola, where the curve turns, is a significant feature, and the \(x\)-intercepts found by solving the equation mark where the parabola crosses the \(x\)-axis. For the equation \(5x^2 + x - 18= 0\), after factoring and solving for \(x\), we would plot the points \((9/5, 0)\) and \((-2, 0)\) along with other points obtained by selecting values for \(x\), and then draw the parabola through these points.

Quadratic Intercepts

Quadratic intercepts are the points at which a quadratic function crosses the axes. There are two types of intercepts: \(x\)-intercepts and the \(y\)-intercept. The \(x\)-intercepts, also known as zeros or roots, are found by solving the quadratic equation. These intercepts are essential for graphing and understanding the function's behavior.

The \(y\)-intercept, on the other hand, occurs where the graph crosses the \(y\)-axis, which is when \(x=0\). For the equation given in our exercise \(5x^2 + x - 18= 0\), we already determined the \(x\)-intercepts to be \(x = 9/5\) and \(x = -2\). The \(y\)-intercept can be found by substituting \(x=0\) into the equation, giving \(y = -18\). These intercepts are key to sketching the graph and provide insight into the function, such as indicating the potential maximum or minimum values based on the parabola's direction.