Zeros/Roots of a Polynomial - College Algebra

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$\begin{align}&$\text{Find the value(s) of x where the function: $}$f(x)=322x^{2}$ - 325x + 75\&$\text{has a zero value.}$\end{align}$

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Answer

$\begin{align}&$\text{Given $}$f(x)=322x^{2}$ - 325x + 75$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&(14x - 5)\cdot (23x - 15)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-325)\pm\sqrt{(-325)^2$-4(322)(75)}$}{2(322)}=\frac{325\pm95}{644}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$ \ x=\frac{5}{14}$$\text{ and }$$\frac{15}{23}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: $}$f(x)=140x^{2}$ - 548x - 168\&$\text{crosses the x-axis.}$\end{align}$

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$\begin{align}&$\text{The function $}$f(x)=140x^{2}$ - 548x - 168$\text{ crosses the x-axis}$\&$\text{when it has a value of zero. To find the x values that}$\&$\text{satisfy this, we can either factor the equation:}$\&4\cdot (7x + 2)\cdot (5x - 21)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-548)\pm\sqrt{(-548)^2$-4(140)(-168)}$}{2(140)}=\frac{548\pm628}{280}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=-$\frac{2}{7}$$\text{ and }$$\frac{21}{5}$.\end{align}$

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$\begin{align}&$\text{Find the root(s) the function: $}$f(x)=270x^{2}$ - 596x - 448\end{align}$

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$\begin{align}&$\text{The roots of the function $}$f(x)=270x^{2}$ - 596x - 448$\text{ are the x values}$\&$\text{that give it a value of zero. To find these x values}$\&$\text{we can either factor the equation:}$\&2\cdot (27x + 16)\cdot (5x - 14)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-596)\pm\sqrt{(-596)^2$-4(270)(-448)}$}{2(270)}=\frac{596\pm916}{540}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=\frac{14}{5}$$\text{ and }$-$\frac{16}{27}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=123x - $460x^{2}$ + 532\&$\text{has a zero value.}$\end{align}$

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$\begin{align}&$\text{Given }$f(x)=123x - $460x^{2}$ + 532$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&-(20x + 19)\cdot (23x - 28)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(123)\pm\sqrt{(123)^2$-4(-460)(532)}$}{2(-460)}=\frac{-123\pm997}{-920}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=\frac{28}{23}$$\text{ and }$-$\frac{19}{20}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: $}$f(x)=100x^{2}$ - 627x - 324\&$\text{crosses the x-axis.}$\end{align}$

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$\begin{align}&$\text{The function $}$f(x)=100x^{2}$ - 627x - 324$\text{ crosses the x-axis}$\&$\text{when it has a value of zero. To find the x values that}$\&$\text{satisfy this, we can either factor the equation:}$\&(25x + 12)\cdot (4x - 27)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-627)\pm\sqrt{(-627)^2$-4(100)(-324)}$}{2(100)}=\frac{627\pm723}{200}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$ \ x=\frac{27}{4}$$\text{ and }$-$\frac{12}{25}$.\end{align}$

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$\begin{align}&$\text{Find the root(s) the function: }$f(x)=771x - $504x^{2}$ - 276\end{align}$

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$\begin{align}&$\text{The roots of the function }$f(x)=771x - $504x^{2}$ - 276$\text{ are the x values}$\&$\text{that give it a value of zero. To find these x values}$\&$\text{we can either factor the equation:}$\&-3\cdot (7x - 4)\cdot (24x - 23)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(771)\pm\sqrt{(771)^2$-4(-504)(-276)}$}{2(-504)}=\frac{-771\pm195}{-1008}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=\frac{4}{7}$$\text{ and }$$\frac{23}{24}$.\end{align}$

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$\begin{align}&$\text{Find the root(s) the function: }$f(x)=293x - $75x^{2}$ - 66\end{align}$

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$\begin{align}&$\text{The roots of the function }$f(x)=293x - $75x^{2}$ - 66$\text{ are the x values}$\&$\text{that give it a value of zero. To find these x values}$\&$\text{we can either factor the equation:}$\&-(25x - 6)\cdot (3x - 11)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(293)\pm\sqrt{(293)^2$-4(-75)(-66)}$}{2(-75)}=\frac{-293\pm257}{-150}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=\frac{6}{25}$$\text{ and }$$\frac{11}{3}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=350x + $112x^{2}$ + 42\&$\text{crosses the x-axis.}$\end{align}$

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$\begin{align}&$\text{The function }$f(x)=350x + $112x^{2}$ + 42$\text{ crosses the x-axis}$\&$\text{when it has a value of zero. To find the x values that}$\&$\text{satisfy this, we can either factor the equation:}$\&14\cdot (x + 3)\cdot (8x + 1)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(350)\pm\sqrt{(350)^2$-4(112)(42)}$}{2(112)}=\frac{-350\pm322}{224}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=-3$\text{ and }$-$\frac{1}{8}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=- 197x - $532x^{2}$ - 15\&$\text{has a zero value.}$\end{align}$

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Answer

$\begin{align}&$\text{Given }$f(x)=- 197x - $532x^{2}$ - 15$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&-(19x + 5)\cdot (28x + 3)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-197)\pm\sqrt{(-197)^2$-4(-532)(-15)}$}{2(-532)}=\frac{197\pm83}{-1064}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=-$\frac{5}{19}$$\text{ and }$-$\frac{3}{28}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=24 - $900x^{2}$ - 300x\&$\text{has a zero value.}$\end{align}$

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$\begin{align}&$\text{Given }$f(x)=24 - $900x^{2}$ - 300x$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&-12\cdot (5x + 2)\cdot (15x - 1)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-300)\pm\sqrt{(-300)^2$-4(-900)(24)}$}{2(-900)}=\frac{300\pm420}{-1800}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=-$\frac{2}{5}$$\text{ and }$$\frac{1}{15}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=- 295x - $250x^{2}$ - 66\&$\text{has a zero value.}$\end{align}$

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$\begin{align}&$\text{Given }$f(x)=- 295x - $250x^{2}$ - 66$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&-(25x + 22)\cdot (10x + 3)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-295)\pm\sqrt{(-295)^2$-4(-250)(-66)}$}{2(-250)}=\frac{295\pm145}{-500}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=-$\frac{3}{10}$$\text{ and }$-$\frac{22}{25}$.\end{align}$

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Question

$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=15x + $90x^{2}$ - 5\&$\text{has a zero value.}$\end{align}$

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$\begin{align}&$\text{Given }$f(x)=15x + $90x^{2}$ - 5$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&5\cdot (3x + 1)\cdot (6x - 1)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(15)\pm\sqrt{(15)^2$-4(90)(-5)}$}{2(90)}=\frac{-15\pm45}{180}$\&$\text{Either way, we find thatthe function crosses the x-axis at}$\ x=\frac{1}{6}$$\text{ and }$-$\frac{1}{3}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=279x - $51x^{2}$ + 162\&$\text{has a zero value.}$\end{align}$

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$\begin{align}&$\text{Given }$f(x)=279x - $51x^{2}$ + 162$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&-3\cdot (17x + 9)\cdot (x - 6)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(279)\pm\sqrt{(279)^2$-4(-51)(162)}$}{2(-51)}=\frac{-279\pm333}{-102}$\&$\text{Either way, we find that the function crosses the x-axis at}$ \ x=6$\text{ and }$-$\frac{9}{17}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=153x - $380x^{2}$ - 7\&$\text{has a zero value.}$\end{align}$

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Answer

$\begin{align}&$\text{Given }$f(x)=153x - $380x^{2}$ - 7$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&-(19x - 1)\cdot (20x - 7)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(153)\pm\sqrt{(153)^2$-4(-380)(-7)}$}{2(-380)}=\frac{-153\pm113}{-760}$\&$\text{Either way, we find that the function crosses the x-axis at}$\ x=\frac{1}{19}$$\text{ and }$$\frac{7}{20}$.\end{align}$

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Question

$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=154x + $14x^{2}$ + 252\&$\text{has a zero value.}$\end{align}$

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$\begin{align}&$\text{Given }$f(x)=154x + $14x^{2}$ + 252$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&14\cdot (x + 9)\cdot (x + 2)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(154)\pm\sqrt{(154)^2$-4(14)(252)}$}{2(14)}=\frac{-154\pm98}{28}$\&$\text{Either way, we find that the function crosses the x-axis at}$\ x=-2$\text{ and }$-9.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=46 - $6x^{2}$ - 65x\&$\text{has a zero value.}$\end{align}$

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$\begin{align}&$\text{Given }$f(x)=46 - $6x^{2}$ - 65x$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&-(2x + 23)\cdot (3x - 2)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-65)\pm\sqrt{(-65)^2$-4(-6)(46)}$}{2(-6)}=\frac{65\pm73}{-12}$\&$\text{Either way, we find that the function crosses the x-axis at}$\ x=\frac{2}{3}$$\text{ and }$-$\frac{23}{2}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=221x - $184x^{2}$ + 44\&$\text{has a zero value.}$\end{align}$

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$\begin{align}&$\text{Given }$f(x)=221x - $184x^{2}$ + 44$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&-(23x + 4)\cdot (8x - 11)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(221)\pm\sqrt{(221)^2$-4(-184)(44)}$}{2(-184)}=\frac{-221\pm285}{-368}$\&$\text{Either way, we find that the function crosses the x-axis at}$ \ x=-$\frac{4}{23}$$\text{ and }$$\frac{11}{8}$.\end{align}$

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$\begin{align}&$\text{Find the root(s) the function: $}$f(x)=66x^{2}$ - 248x - 280\end{align}$

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Answer

$\begin{align}&$\text{The roots of the function $}$f(x)=66x^{2}$ - 248x - 280$\text{ are the x values}$\&$\text{that give it a value of zero. To find these x values}$\&$\text{we can either factor the equation:}$\&2\cdot (11x + 10)\cdot (3x - 14)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(-248)\pm\sqrt{(-248)^2$-4(66)(-280)}$}{2(66)}=\frac{248\pm368}{132}$\&$\text{Either way, we find that the function crosses the x-axis at}$\ x=\frac{14}{3}$$\text{ and }$-$\frac{10}{11}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=306x - $270x^{2}$ - 72\&$\text{crosses the x-axis.}$\end{align}$

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Answer

$\begin{align}&$\text{The function }$f(x)=306x - $270x^{2}$ - 72$\text{ crosses the x-axis}$\&$\text{when it has a value of zero. To find the x values that}$\&$\text{satisfy this, we can either factor the equation:}$\&-18\cdot (3x - 1)\cdot (5x - 4)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(306)\pm\sqrt{(306)^2$-4(-270)(-72)}$}{2(-270)}=\frac{-306\pm126}{-540}$\&$\text{Either way, we find that the function crosses the x-axis at}$\ x=\frac{1}{3}$$\text{ and }$$\frac{4}{5}$.\end{align}$

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$\begin{align}&$\text{Find the value(s) of x where the function: }$f(x)=232x + $120x^{2}$ - 208\&$\text{has a zero value.}$\end{align}$

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Answer

$\begin{align}&$\text{Given }$f(x)=232x + $120x^{2}$ - 208$\text{, to find when it has a value of}$\&$\text{zero, we need a method to determine the x values that}$\&$\text{make this happen. We can either factor the equation:}$\&8\cdot (5x + 13)\cdot (3x - 2)\&$\text{or, if this is not intuitive, use the quadratic $formula:}$\&$\frac{-b\pm\sqrt{b^2$-4ac}$}{2a}($\text{for equations of the form: $}$a^2x$$+bx+c)\&$\frac{-(232)\pm\sqrt{(232)^2$-4(120)(-208)}$}{2(120)}=\frac{-232\pm392}{240}$\&$\text{Either way, we find that the function crosses the x-axis at}$ \ x=\frac{2}{3}$$\text{ and }$-$\frac{13}{5}$.\end{align}$

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