# What are the x-values of the solutions of this system? [latex] \left \{ {{y=x+3} \atop {y=2x^2-3}} \right. [/latex]

so since they are equal x+3=y=2x^2-3 x+3=2x^2-3 make one side 0 subtract (x+3) ffrom both sides 0=2x^2-x-6 factor 0=(2x+3)(x-2) se each to zero 2x+3=0 x-2=0 2x+3=0 subtract 3 2x=-3 divide 2 x=-3/2 x-2=0 add 2 x=2 x=2 or x=-3/2

So, We notice that both equations say that "y" is equal to something. Since y is constant, we can assume that both of the other sides are equal, since they are both equal to y. First, we can set the sides equal to each other. [latex]x+3=2x^2-3[/latex] Now we can solve for x. Subtract x from both sides. [latex]3=2x^2-x-3[/latex] Subtract 3 from both sides. [latex]0=2x^2-x-6\ or\ 2x^2-x-6=0[/latex] We can now factor. [latex]2x^2-x-6=0--\ \textgreater \ (x-2)(2x+3)=0[/latex] Set each factor equal to zero. x - 2 = 0 Add 2 to both sides. x = 2 2x + 3 = 0 Subtract 3 from both sides. 2x = -3 Divide both sides by 2. [latex]x= \frac{-3}{2} [/latex] We can now substitute 2 for x in the first equation and see what we get for y. y = 2 + 3 y = 5 So one pair of solutions is (2,5). Now, let's see what we get if we use the second value for x. [latex]y= \frac{-3}{2} +3[/latex] [latex]y= \frac{-3}{2} + \frac{6}{2} [/latex] [latex]y= \frac{3}{2}[/latex] So another ordered pair is [latex]( \frac{-3}{2} , \frac{3}{2} )[/latex] Let's check our solutions. (2,5) We already know this works in the first equation, so we need to test the second equation. [latex]5=2(2)^2-3[/latex] 5 = 2(4) - 3 5 = 8 - 3 5 = 5 This checks. Now, let's check the second solution. We already know this works in the first equation, as we used that equation to find the y value. We just need to test the second equation. [latex] \frac{3}{2} = 2(-\frac{3}{2}) ^2-3[/latex] [latex] \frac{3}{2} = 2( \frac{9}{4}) -3[/latex] [latex] \frac{3}{2} = \frac{9}{2} -3[/latex] [latex] \frac{3}{2} = \frac{9}{2} - \frac{6}{2} [/latex] [latex] \frac{3}{2} = \frac{3}{2} [/latex] This also checks. Therefore, our solutions are: S = {[latex](2,5),( \frac{3}{2} ,- \frac{3}{2} )[/latex]}