Use factoring to determine how many times the graph of each function intersects the $x$-axis. Identify each zero.
30. $y=x^2-8 x+16$
31. $y=x^2+4 x+4$
32. $y=x^2+2 x-24$
33. $y=x^2+12 x+32$

Factor each quadratic equation.
Solve for x to find the zeros.
Count the number of distinct real roots to determine the number of x-intercepts.
The zeros are: x = 4 , x = − 2 , x = − 6 and x = 4 , x = − 4 and x = − 8 . The number of intersections are 1, 1, 2, 2 respectively. x = 4 , x = − 2 , x = − 6 , 4 , x = − 4 , − 8 ​

Explanation

Problem Analysis We are given four quadratic functions and asked to find how many times each graph intersects the x-axis and to identify the zeros. To do this, we will factor each quadratic, set it equal to zero, and solve for x. The number of real solutions will tell us how many times the graph intersects the x-axis.

Solving Function 1 For the first function, y = x 2 − 8 x + 16 , we set y = 0 and factor the quadratic expression: x 2 − 8 x + 16 = 0
( x − 4 ) ( x − 4 ) = 0
( x − 4 ) 2 = 0
So, x = 4 . This quadratic has one real root, x = 4 . Therefore, the graph intersects the x-axis once at x = 4 .

Solving Function 2 For the second function, y = x 2 + 4 x + 4 , we set y = 0 and factor the quadratic expression: x 2 + 4 x + 4 = 0
( x + 2 ) ( x + 2 ) = 0
( x + 2 ) 2 = 0
So, x = − 2 . This quadratic has one real root, x = − 2 . Therefore, the graph intersects the x-axis once at x = − 2 .

Solving Function 3 For the third function, y = x 2 + 2 x − 24 , we set y = 0 and factor the quadratic expression: x 2 + 2 x − 24 = 0
( x + 6 ) ( x − 4 ) = 0
So, x = − 6 or x = 4 . This quadratic has two real roots, x = − 6 and x = 4 . Therefore, the graph intersects the x-axis twice, at x = − 6 and x = 4 .

Solving Function 4 For the fourth function, y = x 2 + 12 x + 32 , we set y = 0 and factor the quadratic expression: x 2 + 12 x + 32 = 0
( x + 4 ) ( x + 8 ) = 0
So, x = − 4 or x = − 8 . This quadratic has two real roots, x = − 4 and x = − 8 . Therefore, the graph intersects the x-axis twice, at x = − 4 and x = − 8 .

Final Answer In summary:



y = x 2 − 8 x + 16 intersects the x-axis once at x = 4 .
y = x 2 + 4 x + 4 intersects the x-axis once at x = − 2 .
y = x 2 + 2 x − 24 intersects the x-axis twice at x = − 6 and x = 4 .
y = x 2 + 12 x + 32 intersects the x-axis twice at x = − 4 and x = − 8 .

Examples
Understanding the x-intercepts of a quadratic function is crucial in various real-world applications. For example, in physics, the height of a projectile over time can be modeled by a quadratic function. The x-intercepts would represent the times when the projectile is at ground level. Similarly, in business, a quadratic function can model the profit of a company as a function of the price of a product. The x-intercepts would represent the break-even points where the company neither makes a profit nor incurs a loss. Factoring quadratics allows us to find these critical points and make informed decisions.

Answered by GinnyAnswer | 2025-07-03

The functions intersect the x-axis at the following points: Function 30 at x = 4 , Function 31 at x = − 2 , Function 32 at x = − 6 and x = 4 , and Function 33 at x = − 4 and x = − 8 . The number of x-intercepts are 1, 1, 2, and 2 respectively.
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Answered by Anonymous | 2025-07-04