The Solutions Formula for Quadratic Equations

$$f(x) = 3x^2 + 8x - 8$$ $$\displaystyle g(x) = 2x^2 +2x +8$$ For what values of _x_ do the functions above have the same value?

Incorrect. [[snippet]] When _x_ is –3, the two functions do not have the same value. $$\displaystyle f(-3) = 3(-3)^2 + 8(-3) - 8 = 27 - 24 - 8 = -5$$ $$\displaystyle g(-3) = 2(-3)^2 +2(-3) +8 = 18 - 6 + 8 = 20$$ $$\displaystyle 20 \neq -5$$

Incorrect. [[snippet]] If you __Reverse Plug In__ 8, the functions will not be equal. $$\displaystyle f(8) = 3(8)^2 + 8(8) - 8 = 192 + 64 - 8 = 148$$ $$\displaystyle g(8) = 2(8)^2 +2(8) +8 = 128 + 16 + 8 = 152$$ $$\displaystyle 152 \neq 148$$

As a third option, you can __Reverse Plug In__ the choices and evaluate the functions to find the correct values. $$\displaystyle f(2) = 3(2)^2 + 8(2) - 8 = 12 + 16 - 8 = 20$$ $$\displaystyle g(2) = 2(2)^2 +2(2) +8 = 20$$ $$\displaystyle f(-8) = 3(-8)^2 + 8(-8) - 8 = 192 - 64 - 8 = 120$$ $$\displaystyle g(-8) = 2(-8)^2 +2(-8) +8 = 128 - 16 + 8 = 120$$

Incorrect. [[snippet]] The two functions are not the same when you __Reverse Plug In__ _x_ = 3. $$\displaystyle f(3) = 3(3)^2 + 8(3) - 8 = 27 + 24 - 8 = 43$$ $$\displaystyle g(3) = 2(3)^2 +2(3) +8 = 18 + 6 + 8 = 32$$

Incorrect. [[snippet]] If you __Reverse Plug In__ 4 for _x_, the two functions are not equal. $$\displaystyle f(4) = 3(4)^2 + 8(4) - 8 = 72$$ $$\displaystyle g(4) = 2(4)^2 +2(4) +8 = 48 $$

Correct. You can set the right side of both equations equal to one another. $$\displaystyle 3x^2 + 8x - 8= 2x^2 +2x +8$$ Since you are dealing with quadratic equations, simplify the new equation to be equal to 0 and use the solutions formula for quadratic equations. $$\displaystyle x^2 + 6x - 16 = 0$$ $$\displaystyle x_{1, 2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-6 \pm \sqrt{6^2 + 4(16}}{2}$$ $$\displaystyle x_{1, 2} = \frac{-6 \pm \sqrt{36 + 64}}{2} = \frac{-6 \pm \sqrt{100}}{2} = \frac{-6 \pm 10}{2}$$ $$\displaystyle x = 2, -8$$

Alternatively, you can factor the expression that is equal to 0. $$\displaystyle x^2 + 6x - 16 = 0$$ $$\displaystyle (x - 2)(x + 8) = 0$$ $$\displaystyle x = 2, -8$$

_x_ = –3, –8

_x_ = –2, 8

_x_ = 2, –8

_x_ = 2, 3

_x_ = 2, 4

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