(f)/(g)(x)=(11 x+2x^(2))/(-7x-3x^(2)+4) Find the domain

Define Domain: To find the domain of the function (f)/(g)(x)=(11x+2x^2)/(-7x-3x^2+4), we need to determine the values of x for which the function is defined. The only restriction for the domain comes from the denominator, as it cannot be equal to zero because division by zero is undefined.Set Denominator to Zero: Set the denominator equal to zero and solve for x to find the values that are not in the domain. -7x - 3x^2 + 4 = 0Rearrange Quadratic Equation: Rearrange the equation to standard quadratic form. 3x^2 + 7x - 4 = 0Factor or Use Quadratic Formula: Factor the quadratic equation, if possible, to find the roots. If factoring is difficult or not possible, use the quadratic formula. However, this quadratic does not factor nicely, so we will use the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a), where a = 3, b = 7, and c = -4.Substitute Values into Formula: Substitute the values of a, b, and c into the quadratic formula. x = [-7 ± sqrt(7^2 - 4(3)(-4))] / (2(3)) x = [-7 ± sqrt(49 + 48)] / 6 x = [-7 ± sqrt(97)] / 6Calculate Discriminant and Roots: Calculate the discriminant (inside the square root) and the roots. x = [-7 ± sqrt(97)] / 6 This gives us two distinct real roots.Identify Domain: The roots are the values of x that make the denominator zero, so they are not included in the domain. Therefore, the domain of the function is all real numbers except the two roots we found.

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$\frac{f}{g}(x) = \frac{11x + 2x^2}{-7x - 3x^2 + 4}$ Find the domain

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Algebra 1

Domain and range of quadratic functions: equations

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\frac{f}{g}(x) = \frac{11x + 2x^2}{-7x - 3x^2 + 4}

Find the domain

Define Domain: To find the domain of the function $\frac{f}{g}(x)=\frac{11x+2x^2}{-7x-3x^2+4}$ , we need to determine the values of $x$ for which the function is defined. The only restriction for the domain comes from the denominator, as it cannot be equal to zero because division by zero is undefined.
Set Denominator to Zero: Set the denominator equal to zero and solve for $x$ to find the values that are not in the domain. $\newline$ $-7x - 3x^2 + 4 = 0$
Rearrange Quadratic Equation: Rearrange the equation to standard quadratic form. $\newline$ $3x^2 + 7x - 4 = 0$
Factor or Use Quadratic Formula: Factor the quadratic equation, if possible, to find the roots. If factoring is difficult or not possible, use the quadratic formula. $\newline$ However, this quadratic does not factor nicely, so we will use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 3$ , $b = 7$ , and $c = -4$ .
Substitute Values into Formula: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula. $\newline$ $x = \frac{-7 \pm \sqrt{7^2 - 4(3)(-4)}}{2(3)}$ $\newline$ $x = \frac{-7 \pm \sqrt{49 + 48}}{6}$ $\newline$ $x = \frac{-7 \pm \sqrt{97}}{6}$
Calculate Discriminant and Roots: Calculate the discriminant (inside the square root) and the roots. $\newline$ $x = \frac{-7 \pm \sqrt{97}}{6}$ $\newline$ This gives us two distinct real roots.
Identify Domain: The roots are the values of $x$ that make the denominator zero, so they are not included in the domain. Therefore, the domain of the function is all real numbers except the two roots we found.