A product equal to zero gives you a quick way to find solutions without expanding it. An x-intercept is where a graph meets the horizontal axis, so graph the product in Desmos and look for intercepts. Then match the relevant value to a choice. Check the sign of a fraction by testing it in a factor; the number written in a factor is not automatically the solution.
Hints
- Hint 1
At an x-intercept, the graph’s height is . Graph the entire product as written, then look for the points where it meets the horizontal axis.
- Hint 2
Desmos may label a fractional intercept with a rounded decimal. Match it to a choice, then check the exact fraction in one factor. Which factor takes less work to test?
Step-by-step
Approach 1: Graph the product in Desmos
- Step 1
Graph the left side
Type in Desmos. Keep the parentheses: the equation asks where this whole product has a value of , so you’re looking for where its graph meets the horizontal axis.
- Step 2
Read the solutions from the graph
Add for the right side of the equation, then click all three intersections. Desmos shows , approximately , and . Their -coordinates are solutions; the is the value of both sides, not another solution.
- Step 3
Check the exact fraction
The intercept near points to . To check that exact value, type . Desmos prints , so the factor is zero and the original product is zero. The solution among the choices is . Choice D.
Approach 2: Use the zero-product property
- Step 1
Split the product
The zero-product property says that when a product is , at least one factor must be . So either factor can give a solution:
- Step 2
Isolate the square
For the first factor, add to both sides:
- Step 3
Find both square roots
Both and square to , so this factor gives two solutions:
Neither is listed among the choices.
- Step 4
Isolate the term with x
Use the other factor. Subtract from both sides of :
- Step 5
Solve the linear factor
Divide both sides by :
The negative sign stays with the , so the listed solution is . Choice D.