Solve for x:
$(2x+3)(x-5)=0$
$(2x+3)(x-5)=0$
Solve for x:
$(2x+3)(x-5)=0$
The equation provided is \((2x + 3)(x - 5) = 0\). This is a quadratic equation already presented in its factored form. To find the values of \(x\) that satisfy this equation, we use the Zero Product Property, which states that if the product of two or more factors is zero, then at least one of the factors must itself be equal to zero. This allows us to break the quadratic equation down into two simpler linear equations.
Based on the Zero Product Property, we can set each factor equal to zero:
Now, we solve each linear equation individually:
For the first equation:
\(2x + 3 = 0\)
Subtract 3 from both sides:
\(2x = -3\)
Divide by 2:
\(x = -\frac{3}{2} = -1.5\)
For the second equation:
\(x - 5 = 0\)
Add 5 to both sides:
\(x = 5\)
The values that satisfy the equation \((2x + 3)(x - 5) = 0\) are \(x = -1.5\) and \(x = 5\). These are the roots of the quadratic equation. Substituting either value back into the original expression results in a product of zero, confirming our solutions. Thus, the solution set for \(x\) is \(\{-1.5, 5\}\).
Imagine you are designing a trajectory for a projectile or analyzing the profit margins of a small business based on production units. In these cases, you often arrive at a quadratic equation that needs to be solved to find the 'break-even' points where the value is zero. By identifying these points, such as in the equation \((2x + 3)(x - 5) = 0\), you can determine the specific conditions (like the number of units or time) that result in a zero outcome, helping you make informed decisions in real-world engineering or financial modeling.