Question 62·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

0 / 233

$(x^2 - 9)(3x + 5) = 0$

Which of the following is a solution to the given equation?

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 $0$ . 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 $y=(x^2-9)(3x+5)$ in Desmos. Keep the parentheses: the equation asks where this whole product has a value of $0$ , so you’re looking for where its graph meets the horizontal axis.
Step 2
Read the solutions from the graph
Add $y=0$ for the right side of the equation, then click all three intersections. Desmos shows $(-3,0)$ , approximately $(-1.667,0)$ , and $(3,0)$ . Their $x$ -coordinates are solutions; the $0$ is the value of both sides, not another solution.
Step 3
Check the exact fraction
The intercept near $-1.667$ points to $-\frac{5}{3}$ . To check that exact value, type $3\left(-\frac{5}{3}\right)+5$ . Desmos prints $0$ , so the factor $3x+5$ is zero and the original product is zero. The solution among the choices is $-\frac{5}{3}$ . Choice D.

Approach 2: Use the zero-product property
Step 1
Split the product
The zero-product property says that when a product is $0$ , at least one factor must be $0$ . So either factor can give a solution:
$x^2-9=0\quad\text{or}\quad 3x+5=0.$
Step 2
Isolate the square
For the first factor, add $9$ to both sides:
$x^2=9.$
Step 3
Find both square roots
Both $3$ and $-3$ square to $9$ , so this factor gives two solutions:
$x=3\quad\text{or}\quad x=-3.$
Neither is listed among the choices.
Step 4
Isolate the term with x
Use the other factor. Subtract $5$ from both sides of $3x+5=0$ :
$3x=-5.$
Step 5
Solve the linear factor
Divide both sides by $3$ :
$x=-\frac{5}{3}.$
The negative sign stays with the $5$ , so the listed solution is $-\frac{5}{3}$ . Choice D.

Question 62·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this