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$y = - \dfrac { 1 } { 3 } x ^ { 2 }$
$y = 9$
$x$Intercept
$\left ( 0 , 0 \right )$
$y$Intercept
$\left ( 0 , 0 \right )$
Maximum
$\left ( 0 , 0 \right )$
Standard form
$y = - \dfrac { 1 } { 3 } x ^ { 2 }$
$- \dfrac{ 1 }{ 3 } x ^{ 2 } = 9$
$\begin{array} {l} x = 3 \sqrt{ 3 } i \\ x = - 3 \sqrt{ 3 } i \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = 9$
 Calculate the multiplication expression 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = 9$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = \color{#FF6800}{ 9 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 27 }$
$- x ^ { 2 } = \color{#FF6800}{ 27 }$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 }$
$x ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = 0$
 Move the constant to the right side and change the sign 
$x ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 27 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ - } \color{#FF6800}{ 27 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 27 } }$
$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 27 } }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i } \end{array}$
$\begin{array} {l} x = 3 \sqrt{ 3 } i \\ x = - 3 \sqrt{ 3 } i \end{array}$
Calculate using the quodratic formula$($Imaginary root solution$)$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = 9 $Calculate the multiplication expression$ \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = 9\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = \color{#FF6800}{ 9 } $Multiply both sides by the least common multiple for the denominators to eliminate the fraction$ \color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 27 }- x ^ { 2 } = \color{#FF6800}{ 27 } $Move the expression to the left side and change the symbol$ - x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = 0\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 } $Change the symbols of both sides of the equation$ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 }\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 } $Solve the quadratic equation$ ax^{2}+bx+c=0 $using the quadratic formula$ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 0 \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 27 } } { 2 \times 1 } }x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 27 } } { 2 \times 1 } $0 has no sign$ x = \dfrac { \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 27 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 27 } } { 2 \times 1 } $The power of 0 is 0$ x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times 27 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times 27 } } { 2 \times 1 } $0 does not change when you add or subtract$ x = \dfrac { 0 \pm \sqrt{ - 4 \times 1 \times 27 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 27 } } } { 2 \times 1 } $Organize the part that can be taken out of the radical sign inside the square root symbol$ x = \dfrac { 0 \pm \color{#FF6800}{ 6 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i } } { 2 \times 1 }x = \dfrac { 0 \pm 6 \sqrt{ 3 } i } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } } $Multiplying any number by 1 does not change the value$ x = \dfrac { 0 \pm 6 \sqrt{ 3 } i } { \color{#FF6800}{ 2 } }\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm 6 \sqrt{ 3 } i } { 2 } } $Separate the answer$ \begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 + 6 \sqrt{ 3 } i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 - 6 \sqrt{ 3 } i } { 2 } } \end{array}\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 + 6 \sqrt{ 3 } i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 - 6 \sqrt{ 3 } i } { 2 } } \end{array} $Organize the expression$ \begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \sqrt{ 3 } i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 6 \sqrt{ 3 } i } { 2 } } \end{array}\begin{array} {l} x = \dfrac { 6 \sqrt{ 3 } i } { 2 } \\ x = \color{#FF6800}{ \dfrac { - 6 \sqrt{ 3 } i } { 2 } } \end{array} $Move the minus sign to the front of the fraction$ \begin{array} {l} x = \dfrac { 6 \sqrt{ 3 } i } { 2 } \\ x = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 6 \sqrt{ 3 } i } { 2 } } \end{array}\begin{array} {l} x = \color{#FF6800}{ \dfrac { 6 \sqrt{ 3 } i } { 2 } } \\ x = - \dfrac { 6 \sqrt{ 3 } i } { 2 } \end{array} $Reduce the fraction$ \begin{array} {l} x = \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i } \\ x = - \dfrac { 6 \sqrt{ 3 } i } { 2 } \end{array}\begin{array} {l} x = 3 \sqrt{ 3 } i \\ x = - \color{#FF6800}{ \dfrac { 6 \sqrt{ 3 } i } { 2 } } \end{array} $Reduce the fraction$ \begin{array} {l} x = 3 \sqrt{ 3 } i \\ x = - \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i } \right ) \end{array}\begin{array} {l} x = 3 \sqrt{ 3 } i \\ x = \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i } \right ) \end{array} $Get rid of unnecessary parentheses$ \begin{array} {l} x = 3 \sqrt{ 3 } i \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ i } \end{array} $Do not have the solution$ $Calculate using the quadratic formula$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = 9 $Calculate the multiplication expression$ \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = 9\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = \color{#FF6800}{ 9 } $Multiply both sides by the least common multiple for the denominators to eliminate the fraction$ \color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 27 }- x ^ { 2 } = \color{#FF6800}{ 27 } $Move the expression to the left side and change the symbol$ - x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = 0\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 } $Change the symbols of both sides of the equation$ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 }x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 27 } } { 2 \times 1 } $0 has no sign$ x = \dfrac { \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 27 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 27 } } { 2 \times 1 } $The power of 0 is 0$ x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times 27 } } { 2 \times 1 }\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm \sqrt{ 0 - 4 \times 1 \times 27 } } { 2 \times 1 } } $Organize the expression$ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm \sqrt{ - 108 } } { 2 } }\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm \sqrt{ - 108 } } { 2 } } $The square root of a negative number does not exist within the set of real numbers$  $Do not have the solution$  $Do not have the real root$ $Find the number of solutions$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = 9 $Calculate the multiplication expression$ \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = 9\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = \color{#FF6800}{ 9 } $Multiply both sides by the least common multiple for the denominators to eliminate the fraction$ \color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 27 }- x ^ { 2 } = \color{#FF6800}{ 27 } $Move the expression to the left side and change the symbol$ - x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = 0\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 } $Change the symbols of both sides of the equation$ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 }\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 } $Determine the number of roots using discriminant,$ D=b^{2}-4ac $from quadratic equation,$ ax^{2}+bx+c=0\color{#FF6800}{ D } = \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 27 }D = \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 27 $The power of 0 is 0$ D = \color{#FF6800}{ 0 } - 4 \times 1 \times 27D = \color{#FF6800}{ 0 } - 4 \times 1 \times 27 $0 does not change when you add or subtract$ D = - 4 \times 1 \times 27D = - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times 27 $Multiplying any number by 1 does not change the value$ D = - 4 \times 27D = \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 27 } $Multiply$ - 4 $and$ 27D = \color{#FF6800}{ - } \color{#FF6800}{ 108 }\color{#FF6800}{ D } = \color{#FF6800}{ - } \color{#FF6800}{ 108 } $Since$ D<0 $, there is no real root of the following quadratic equation$  $Do not have the real root$ \alpha + \beta = 0 , \alpha \beta = 27$Find the sum and product of the two roots of the quadratic equation$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = 9 $Calculate the multiplication expression$ \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = 9\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } } { 3 } } = \color{#FF6800}{ 9 } $Multiply both sides by the least common multiple for the denominators to eliminate the fraction$ \color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 27 }- x ^ { 2 } = \color{#FF6800}{ 27 } $Move the expression to the left side and change the symbol$ - x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = 0\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 } $Change the symbols of both sides of the equation$ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 }\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 27 } = \color{#FF6800}{ 0 } $In the quadratic equation$ ax^{2}+bx+c=0 $, if the two roots are$ \alpha, \beta $, then it is$ \alpha + \beta =-\dfrac{b}{a} $,$ \alpha\times\beta=\dfrac{c}{a}\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 0 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 27 } { 1 } }\alpha + \beta = - \dfrac { 0 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { 27 } { 1 } $If the denominator is 1, the denominator can be removed$ \alpha + \beta = - \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { 27 } { 1 }\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { 27 } { 1 } $0 has no sign$ \alpha + \beta = \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { 27 } { 1 }\alpha + \beta = 0 , \alpha \beta = \dfrac { 27 } { \color{#FF6800}{ 1 } } $If the denominator is 1, the denominator can be removed$ \alpha + \beta = 0 , \alpha \beta = \color{#FF6800}{ 27 } $그래프 보기$ \$
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