Question

Find the zeroes of the polynomial 3x2+4x-4 and verify the relationship between the zeroes and the coefficients of the polynomial.

Answer 1

Heya!!Here is your answer friend ⤵⤵
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$3x {}^{2} + 4x - 4 \\ \\ using \: middle \: term \: splitting \: \\ sum = 4 \\ product = - 12 \\ 3x {}^{2} + 6 x - 2x - 4 \\ 3x(x + 2) - 2(x + 2) \\ (3x - 2)(x + 2) \\ 3x = 2 \: \: and \: x = - 2 \\ x = \frac{2}{3} \: and \: x = - 2 \\ \alpha = \frac{2}{3} \ \: \: \beta = - 2 \\ \\ relationships \: between \: zeroes \: and \: \: cofficients \\ \alpha + \beta = \frac{ - b}{a} \\ - 2 + \frac{2}{3} = \frac{ - 4}{3} \\ \frac{ - 4}{3} = \frac{ - 4}{3} \\ \\ \alpha \beta = \frac{c}{a} \\ \frac{2}{3} \times - 2 = \frac{ - 4}{3} \\ \frac{ - 4}{3} = \frac{ - 4}{3} \\ hence \: verified$
Hope it helps you ✌✌

Answer 2

Answer:

Zeros of a function : are the input value of a function for which the function gives zero as output,

Or we can say that zeroes of a function f(x) are the values for which,

f(x) = 0

Here, the given equation,

$3x^2+4x-4$

Let, $f(x) = 3x^2+4x-4$

For zeroes,

f(x) = 0,

$3x^2+4x-4=0$

$3x^2+6x-2x-4=0$   ( By the middle term splitting )

$3x(x+2)-2(x+2)=0$

$(3x-2)(x+2)=0$

⇒ x = 2/3 or x = -2,

Now, we know that if $\alpha$ and $\beta$ are the zeroes of a quadratic equation then,

$\alpha+\beta=-\frac{\text{Coefficient of x}}{\text{Coefficient of } x^2}$

And,

$\alpha.\beta=\frac{\text{Constant}}{\text{Coefficient of } x^2}$

Here, the coefficient of x² = 3, coefficient of x = 4 and constant term = -4,

Also,

$\frac{2}{3}+(-2)=-\frac{4}{3}$

$\frac{2}{3}\times (-2)=\frac{-4}{3}$

Hence, proved...