Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.
a. [tex]x^2+7 x+9=0[/tex]
b. [tex]x^2-2 x-4=0[/tex]
c. [tex]x^2-4 x+6=0[/tex]

Find the roots of x 2 + 7 x + 9 = 0 . The roots are approximately x = − 5.3 and x = − 1.7 .
Find the roots of x 2 − 2 x − 4 = 0 . The roots are approximately x = − 1.2 and x = 3.2 .
Find the roots of x 2 − 4 x + 6 = 0 . There are no real roots.
The solutions are x ≈ − 5.3 , − 1.7 for (a), x ≈ − 1.2 , 3.2 for (b), and no real roots for (c). x ≈ − 5.3 , − 1.7 ; x ≈ − 1.2 , 3.2 ; no real roots ​

Explanation

Problem Analysis We are given three quadratic equations to solve by graphing: a. x 2 + 7 x + 9 = 0 b. x 2 − 2 x − 4 = 0 c. x 2 − 4 x + 6 = 0 We will find the roots of each equation by finding the x-intercepts of the corresponding quadratic function. If the roots are not integers, we will estimate them to the nearest tenth. We have already found the approximate roots using the tool.

Solving equation a a. For x 2 + 7 x + 9 = 0 , the roots are approximately -5.3 and -1.7.

Solving equation b b. For x 2 − 2 x − 4 = 0 , the roots are approximately -1.2 and 3.2.

Solving equation c c. For x 2 − 4 x + 6 = 0 , there are no real roots since the function has no x-intercepts. The tool confirms that there are no real roots.


Examples
Quadratic equations are useful in many real-world scenarios, such as determining the trajectory of a ball, calculating the area of a garden, or designing suspension bridges. In sports, understanding the trajectory of a ball can help athletes improve their performance. For example, a basketball player can use the equation of a parabola to determine the optimal angle and velocity to shoot the ball for a basket. Similarly, engineers use quadratic equations to design structures that can withstand various forces and stresses.

Answered by GinnyAnswer | 2025-07-03

The approximate roots for the equations are as follows: For x 2 + 7 x + 9 = 0 , roots are x → − 5.3 , − 1.7 ; for x 2 − 2 x − 4 = 0 , roots are x → − 1.2 , 3.2 ; and for x 2 − 4 x + 6 = 0 , there are no real roots.
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Answered by Anonymous | 2025-07-04