Solve the system of equations:

[tex]
\begin{array}{l}
y=2 x^2+12 x-22 \\
y=12 x+10
\end{array}
[/tex]

Write the coordinates in exact form. Simplify all fractions and radicals.

Set the two equations equal: 2 x 2 + 12 x − 22 = 12 x + 10 .
Simplify the equation: 2 x 2 − 32 = 0 , which simplifies further to x 2 − 16 = 0 .
Solve for x by factoring: ( x − 4 ) ( x + 4 ) = 0 , giving x = 4 and x = − 4 .
Substitute x values into y = 12 x + 10 to find corresponding y values: ( 4 , 58 ) and ( − 4 , − 38 ) .

( 4 , 58 ) , ( − 4 , − 38 ) ​
Explanation

Problem Analysis We are given a system of two equations:

y = 2 x 2 + 12 x − 22
y = 12 x + 10
We want to find the solutions (x, y) to this system.

Setting Equations Equal Since both equations are equal to y , we can set them equal to each other:

2 x 2 + 12 x − 22 = 12 x + 10

Simplifying the Equation Now, we simplify the equation by subtracting 12 x and 10 from both sides:

2 x 2 + 12 x − 22 − 12 x − 10 = 0
2 x 2 − 32 = 0

Further Simplification Divide the entire equation by 2:

x 2 − 16 = 0

Solving for x We can solve this quadratic equation by factoring:

( x − 4 ) ( x + 4 ) = 0
So, the solutions for x are x = 4 and x = − 4 .

Solving for y Now we substitute these values of x back into the linear equation y = 12 x + 10 to find the corresponding values of y .

For x = 4 :
y = 12 ( 4 ) + 10 = 48 + 10 = 58
For x = − 4 :
y = 12 ( − 4 ) + 10 = − 48 + 10 = − 38

Final Solutions Therefore, the solutions to the system of equations are ( 4 , 58 ) and ( − 4 , − 38 ) .

Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company's cost function is represented by a quadratic equation and its revenue function is linear, solving the system of equations will give the production levels where cost equals revenue, indicating profitability. Understanding these concepts helps in making informed business decisions and optimizing resource allocation. The solution of the system represents the points where the cost and revenue curves intersect, providing valuable insights into the company's financial performance.

Answered by GinnyAnswer | 2025-07-05

The solutions to the system of equations are the points where the two equations intersect, specifically ( 4 , 58 ) and ( − 4 , − 38 ) . This was found by setting the equations equal, simplifying, and solving the resulting quadratic equation. The corresponding y values were calculated using the linear equation for each x value obtained.
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Answered by Anonymous | 2025-07-08