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Solve the quadratic equation
Answer
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Number of solution
Answer
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Relationship between roots and coefficients
Answer
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Graph
$y = - \dfrac { 1 } { 3 } x ^ { 2 }$
$y = 9$
$x$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 $ $
$D = \color{#FF6800}{ 0 } - 4 \times 1 \times 27$
$D = \color{#FF6800}{ 0 } - 4 \times 1 \times 27$
$ $ 0 does not change when you add or subtract $ $
$D = - 4 \times 1 \times 27$
$D = - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times 27$
$ $ Multiplying any number by 1 does not change the value $ $
$D = - 4 \times 27$
$D = \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 27 }$
$ $ Multiply $ - 4 $ and $ 27$
$D = \color{#FF6800}{ - } \color{#FF6800}{ 108 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ - } \color{#FF6800}{ 108 }$
$ $ 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 $ $
Graph
Solution search results
search-thumbnail-$\left(\dfrac {4} {3}x^{2}2=\right)\times \left(\dfrac {1} {3}x^{2}=x\right)\times \left(-6x$ $z^{2}\right)$
7th-9th grade
Calculus
search-thumbnail-Question $4$ 
If $x$ is $6$ what $|s$ 
$\dfrac {1} {3}x$ 
$1frac\left(1\right)\left(3\right)x$
1st-6th grade
Other
search-thumbnail-Which of the following rational numbers are 
equivalent? 
$0Ptionsy$ 
A \frac{5}{6}, \frac{30}{36} 
B $s\sqrt{rac\left(} -2\right)\left(3\right)\sqrt{1rac} \sqrt{4\right)16\right)4} $ 
C $s\sqrt{11aC\left(} -4\right)1-7b,\sqrt{1rac\left(16\sqrt{35\right)9} } $ 
D \frac{1}{2},\frac{3}{8}
7th-9th grade
Other
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