Question

Solve the equation. $$ (5 x-9.5)^{2}(3 x+6.3)=0 $$

Short answer

The roots of the equation \( (5x - 9.5)^{2}(3x + 6.3) = 0 \) are \( x = 1.9 \) and \( x = -2.1 \).

Step-by-step solution

Understanding and Simplifying the Equation

The equation is \( (5x - 9.5)^{2}(3x + 6.3) = 0 \). This equation can be split into two independent equations \( (5x - 9.5) = 0 \) and \( (3x + 6.3) = 0 \). This is possible due to the zero-product property which states that if \( ab = 0 \), then either \( a = 0 \), \( b = 0 \), or both.

Solve for the root \( x \)

For the equation \( 5x - 9.5 = 0 \), isolate \( x \) by adding 9.5 to both sides and then dividing by 5. This gives \( x = 9.5 / 5 = 1.9 \). Therefore, \( x = 1.9 \) is a root of the equation.

Solve for another root \( x \)

For the equation \( 3x + 6.3 = 0 \), isolate \( x \) by subtracting 6.3 from both sides and then dividing by 3. This gives \( x = -6.3 / 3 = -2.1 \). Therefore, \( x = -2.1 \) is another root of the equation.

Key concepts

Zero-Product Property

The zero-product property is a fundamental principle in algebra that helps in solving quadratic equations. When two or more factors are multiplied together and their product equals zero, \textbf{at least one of the factors must be zero}.

In the given problem, we have an equation \( (5x - 9.5)^2(3x + 6.3) = 0 \). This equation is in fact the product of two terms, \( (5x - 9.5)^2 \) and \( (3x + 6.3) \). Since their product results in zero, it implies that either \( (5x - 9.5)^2 = 0 \) or \( 3x + 6.3 = 0 \) or both. Using the zero-product property enables us to break down complex equations into simpler independent equations that are easier to solve.

Isolate Variable

To find the roots of a quadratic equation, we often need to isolate the variable of interest, typically \( x \).

Imagine you are balancing a scale. To maintain equilibrium, whatever you do to one side of the scale, you must do to the other side. Similarly, with equations, any operation you perform on one side must be performed on the other to maintain equality. When we have \( 5x - 9.5 = 0 \), we add 9.5 to both sides to get \( 5x = 9.5 \) and then divide both sides by 5, yielding \( x = 1.9 \).

By systematically isolating the variable, we simplify the equation step by step, transforming it into a solvable form, which allows us to find the solution efficiently.

Quadratic Roots

Quadratic equations typically have two roots, which are the solutions that satisfy the equation. These roots are the \( x \) values that make the quadratic expression equal to zero. In the context of our example, the quadratic roots are the values of \( x \) that make each factor of the original equation yield a zero product.

From the exercise, we discover one root to be \( x = 1.9 \) by solving \( 5x - 9.5 = 0 \). Similarly, the other root is found by solving the factor \( 3x + 6.3 = 0 \) to get \( x = -2.1 \). Both \( 1.9 \) and \( -2.1 \) are the roots of the given quadratic equation, and they represent the points where the graph of the quadratic expression would intersect the \( x \) axis on a coordinate plane.