How do you evaluate the function $f (x) = 3 x^{2} + 3 x - 2$ $f (x) = 3x^2 + 3x -2$ for $f (\frac{2}{a})$ $f (2/a)$ ?

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How do you evaluate the function $f (x) = 3 x^{2} + 3 x - 2$ $f (x) = 3x^2 + 3x -2$ for $f (\frac{2}{a})$ $f (2/a)$ ?

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Answer 1 · 2015-11-20T21:33:52

By 'evaluate' I assume you mean find $a$ $a$ for $f (\frac{2}{a}) = 0$ $f(2/a)=0$
$a = - 1 \pm \sqrt{7}$ $color(green)(a=-1+- sqrt(7))$

Explanation:

Let $\times y_{1} = 3 x^{2} + 3 x - 2 \to f (x)$ $color(white)(xx)y_1=3x^2+3x-2 ->f(x)$

Then $y_{1} = 3 {(\frac{2}{a})}^{2} + 3 (\frac{2}{a}) - 2 \to f (\frac{2}{a})$ $y_1=3(2/a)^2+3(2/a)-2 -> f(2/a)$

$y_{1} = 3 (\frac{4}{a^{2}}) + 3 (\frac{2}{a}) - 2$ $y_1=3(4/a^2) +3(2/a)-2$

$y_{1} = \frac{12}{a^{2}} + \frac{6}{a} - 2$ $y_1=12/a^2+6/a-2$

Let $\times y_{2} = \frac{1}{y_{1}}$ $color(white)(xx)y_2 =1/y_1$ giving

$y_{2} = \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{1}{2} = 0$ $y_2=1/12a^2+1/6a -1/2 =0$

Multiply throughout by 12 giving:

$y_{2} = a^{2} + 2 a - 6 = 0$ $y_2=a^2+2a-6=0$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Using std form \times x a x^{2} + b x + c = 0$ $color(green)("Using std form"color(white)(xxx) ax^2+bx+c=0)$

$Where \times x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $color(green)("Where" color(white)(xx)x = (-b+-sqrt(b^2-4ac))/(2a)$

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In the context of the given question we have:

$a = \frac{- 2 \pm \sqrt{{(2)}^{2} - 4 (1) (- 6)}}{2 (1)}$ $a=(-2+-sqrt((2)^2-4(1)(-6)))/(2(1))$

$a = \frac{- 2 \pm \sqrt{4 + 24}}{2}$ $a=(-2+-sqrt(4+24))/2$

$a = - 1 \pm \sqrt{7}$ $a=-1+- sqrt(7)$

$a \approx - 1 \pm 2.65$ $a~~ -1+- 2.65$ to 2 decimal places

$a \approx - 3.65 x : x a \approx 1.65$ $a~~-3.65color(white)(x) :color(white)(x) a~~1.65$
Tony B

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How do you evaluate the function $f (x) = 3 x^{2} + 3 x - 2$ $f (x) = 3x^2 + 3x -2$ for $f (\frac{2}{a})$ $f (2/a)$ ?

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Explanation:

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