Certainly! Let's solve the given system of equations step-by-step.
The equations are:
1. [tex]\( y = 2x^2 + 11x - 285 \)[/tex]
2. [tex]\( y = 162x - 54 \)[/tex]
To find where these two equations intersect, we need to set them equal to each other because at the points of intersection, the [tex]\( y \)[/tex]-values and [tex]\( x \)[/tex]-values must be the same for both equations.
So, let's set the two equations equal to each other:
[tex]\[ 2x^2 + 11x - 285 = 162x - 54 \][/tex]
To solve this, we need to move all terms to one side of the equation:
[tex]\[ 2x^2 + 11x - 285 - 162x + 54 = 0 \][/tex]
Combine like terms:
[tex]\[ 2x^2 + 11x - 162x -285 + 54 = 0 \][/tex]
[tex]\[ 2x^2 - 151x - 231 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Now, using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
Here, [tex]\( a = 2 \)[/tex], [tex]\( b = -151 \)[/tex], and [tex]\( c = -231 \)[/tex].
Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = (-151)^2 - 4(2)(-231) \][/tex]
[tex]\[ \Delta = 22801 + 1848 \][/tex]
[tex]\[ \Delta = 24649 \][/tex]
Next, find the roots using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{151 \pm \sqrt{24649}}{4} \][/tex]
[tex]\[ x = \frac{151 \pm 157}{4} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{151 + 157}{4} = \frac{308}{4} = 77 \][/tex]
[tex]\[ x = \frac{151 - 157}{4} = \frac{-6}{4} = -\frac{3}{2} \][/tex]
Now we substitute these [tex]\( x \)[/tex]-values back into either of the original equations to find the corresponding [tex]\( y \)[/tex]-values. We'll use [tex]\( y = 162x - 54 \)[/tex].
For [tex]\( x = 77 \)[/tex]:
[tex]\[ y = 162(77) - 54 \][/tex]
[tex]\[ y = 12474 - 54 \][/tex]
[tex]\[ y = 12420 \][/tex]
For [tex]\( x = -\frac{3}{2} \)[/tex]:
[tex]\[ y = 162\left(-\frac{3}{2}\right) - 54 \][/tex]
[tex]\[ y = -243 - 54 \][/tex]
[tex]\[ y = -297 \][/tex]
So, the solutions to the system of equations are the points:
[tex]\[ (77, 12420) \][/tex]
[tex]\[ \left(-\frac{3}{2}, -297\right) \][/tex]
These solutions represent the points where the graphs of [tex]\( f(x) = 2x^2 + 11x - 285 \)[/tex] and [tex]\( g(x) = 162x - 54 \)[/tex] intersect. Therefore, the correct interpretation of the solutions of the system is:
The points where the graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect.
