Problem 42 - Algebra 2

Question

Solve each equation using any method. When necessary, round real solutions to the nearest hundredth. For imaginary solutions, write exact solutions. $$ 5 x^{2}=210 x $$

Step-by-Step Solution

Answer

The solutions to the equation are x = 0 and x = 42.

Step 1: Simplify the Equation

Start by subtracting 210x from both sides of the equation to move all terms to one side and simplify to standard quadratic form. $$5 x^{2} - 210 x = 0$$

Step 2: Factor Out the Common Factor

Factor out the greatest common factor which is 5x from the left-hand side of the equation. $$5x(x - 42) = 0$$

Step 3: Apply the Zero Product Property

Since the product of two factors is zero, one or both of the factors must be zero. Set each factor equal to zero and solve for x. $$5x = 0$$ $$x - 42 = 0$$

Step 4: Solve for x

Solve the two equations from the previous step. For the first equation: $$5x = 0 \Rightarrow x = 0$$For the second equation: $$x - 42 = 0 \Rightarrow x = 42$$

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental aspect of solving quadratic equations. It essentially states that if the product of two factors is zero, then at least one of the factors must also be zero.

When applied to quadratic equations, this property becomes a powerful tool. After factoring the quadratic equation, we often end up with an expression where two factors are multiplied to equal zero. According to the Zero Product Property, we can then set each factor equal to zero and solve the resulting simpler equations.

For example, in the equation provided, we have the factored form 5x(x - 42) = 0. We can then set each factor to zero: 5x = 0 and x - 42 = 0. Solving these, we find that x could be 0 or 42. The Zero Product Property simplifies the process of finding roots of quadratic equations, turning it into a task of solving basic linear equations.

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations by expressing them as a product of their factors. Factors are numbers or expressions that multiply together to give the original quadratic expression. It is one of the most common methods to solve quadratics, particularly when the roots are rational numbers.

To factor a quadratic equation, first ensure it is in the standard form ax^2 + bx + c = 0. Look for a greatest common factor (GCF) among the terms, and if one exists, factor it out. Next, identify two binomials that multiply to give the original quadratic. These binomials are set to zero and solved for the variable.

In our exercise, the quadratic 5x^2 - 210x = 0 was factored by first taking out the GCF of 5x, leaving us with 5x(x - 42) = 0. This simplifies the solving process and leads directly to applying the Zero Product Property.

Standard Quadratic Form

The standard quadratic form refers to writing a quadratic equation in the format ax^2 + bx + c = 0, where a, b, and c represent real numbers, and a ≠ 0. This standardization is the preliminary step in solving quadratic equations, as it prepares the equation for various solving techniques, including factoring, completing the square, and using the quadratic formula.

By organizing the quadratic equation into this format, we can clearly identify the coefficients and constant term, which are vital for the subsequent factoring process, as seen in the provided exercise. Here, we started by subtracting 210x from both sides to achieve the standard form 5x^2 - 210x = 0. This essential step sets the stage for further action, such as extracting a GCF and employing the Zero Product Property as shown in the steps of the solution.