Problem 23 - 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. $$2 x^{2}=5 x$$

Step-by-Step Solution

Answer

The solutions are \(x=0\) and \(x=2.5\).

Step 1: Re-Writing the Equation

Rearrange the equation to make one side equal to zero. The equation will become \(2x^{2}-5x=0\).

Step 2: Factoring Out Common Term

Factor out the common term which is 'x'. The equation will become \(x(2x - 5)=0\).

Step 3: Apply Zero-Product Property

Now apply Zero-Product Property. If a product of factors equals zero, then at least one of the factors must be zero. Setting each factor equal to zero gives \(x=0\) and \(2x - 5 = 0\).

Step 4: Solve the equation

Solve \(2x - 5 = 0\) for x. We get \(x = 5/2 \) or \(2.5\).

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a fundamental technique used to solve for the roots of quadratic equations, which are equations of the second degree, typically in the form of \(ax^2+bx+c=0\). The process involves breaking down the quadratic polynomial into simpler expressions that can be multiplied together to give the original polynomial. To factor a quadratic equation, you look for two binomials that when multiplied together, give you the quadratic equation.

For example, to factor \(2x^2-5x=0\), we seek two expressions \( (dx + e)\) and \( (fx + g)\) such that \( (dx + e)(fx + g) = 2x^2-5x\). Often, factoring involves trial and error, and recognizing patterns can be helpful. A common factor might be an x, which can be factored out to simplify the equation. In the given problem, the common factor is x, resulting in \( x(2x - 5)=0\) after factoring, which leads us to the next step of applying the zero-product property to find the solution.

Zero-Product Property

The zero-product property is a key principle in algebra which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is very useful when solving quadratic equations that have been factored into binomials.

Using the property, each factor of the product is then set to zero and solved for the variable. For instance, from the factored form \( x(2x - 5)=0\), we can deduce that either \( x = 0\) or \( 2x - 5 = 0\). By solving these simple equations, we can find the values of x that satisfy the original equation. The solutions, known as the roots of the quadratic equation, are the x-coordinates where the graph of the function intersects the x-axis. This property is the bridge between the factored form of the equation and the solutions of the equation.

Graphing Quadratic Functions

Graphing quadratic functions is a powerful way to visualize the solutions to quadratic equations. A quadratic function is typically in the form \(y = ax^2 + bx + c\), and its graph is a parabola. When solving quadratic equations, the points where the parabola intersects the x-axis represent the roots or solutions to the equation.

Graphing can confirm the solutions we find by factoring. For \(2x^2-5x=0\), we would graph \(y = 2x^2 - 5x\) and look for the x-intercepts, which indicate the values of x for which \(y=0\). The intercepts on the graph should correspond with the solutions derived algebraically. This visualization is not only a helpful check but also provides insight into the number of real solutions and the behavior of the function. By graphing, we can see if the parabola opens upwards or downwards and where its vertex lies, adding more layers of understanding to the behavior of quadratic equations.