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$y = 3 x ^ { 2 } + 48$
$y = 0$
$y$Intercept
$\left ( 0 , 48 \right )$
Minimum
$\left ( 0 , 48 \right )$
Standard form
$y = 3 x ^ { 2 } + 48$
$3x ^{ 2 } +48 = 0$
$\begin{array} {l} x = 4 i \\ x = - 4 i \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 48 } = \color{#FF6800}{ 0 }$
 Divide both sides by the coefficient of the leading highest term 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } = \color{#FF6800}{ 0 }$
$x ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 16 } = 0$
 Move the constant to the right side and change the sign 
$x ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 16 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ - } \color{#FF6800}{ 16 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 16 } }$
$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 16 } }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 4 } \color{#FF6800}{ i }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 4 } \color{#FF6800}{ i }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 4 } \color{#FF6800}{ i } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ i } \end{array}$
$\begin{array} {l} x = 4 i \\ x = - 4 i \end{array}$
Calculate using the quodratic formula$($Imaginary root solution$)$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 48 } = \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 16 } } { 2 \times 1 } }x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 16 } } { 2 \times 1 } $0 has no sign$ x = \dfrac { \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 16 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 16 } } { 2 \times 1 } $The power of 0 is 0$ x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times 16 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times 16 } } { 2 \times 1 } $0 does not change when you add or subtract$ x = \dfrac { 0 \pm \sqrt{ - 4 \times 1 \times 16 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } } { 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}{ 8 } \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } { 2 \times 1 }x = \dfrac { 0 \pm 8 \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } { 2 \times 1 } $It is$ \sqrt{-1} = ix = \dfrac { 0 \pm 8 \color{#FF6800}{ i } } { 2 \times 1 }x = \dfrac { 0 \pm 8 i } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } } $Multiplying any number by 1 does not change the value$ x = \dfrac { 0 \pm 8 i } { \color{#FF6800}{ 2 } }\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm 8 i } { 2 } } $Separate the answer$ \begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 + 8 i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 - 8 i } { 2 } } \end{array}\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 + 8 i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 - 8 i } { 2 } } \end{array} $Organize the expression$ \begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 8 i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 8 i } { 2 } } \end{array}\begin{array} {l} x = \dfrac { 8 i } { 2 } \\ x = \color{#FF6800}{ \dfrac { - 8 i } { 2 } } \end{array} $Move the minus sign to the front of the fraction$ \begin{array} {l} x = \dfrac { 8 i } { 2 } \\ x = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 8 i } { 2 } } \end{array}\begin{array} {l} x = \color{#FF6800}{ \dfrac { 8 i } { 2 } } \\ x = - \dfrac { 8 i } { 2 } \end{array} $Reduce the fraction$ \begin{array} {l} x = \color{#FF6800}{ 4 } \color{#FF6800}{ i } \\ x = - \dfrac { 8 i } { 2 } \end{array}\begin{array} {l} x = 4 i \\ x = - \color{#FF6800}{ \dfrac { 8 i } { 2 } } \end{array} $Reduce the fraction$ \begin{array} {l} x = 4 i \\ x = - \left ( \color{#FF6800}{ 4 } \color{#FF6800}{ i } \right ) \end{array}\begin{array} {l} x = 4 i \\ x = \color{#FF6800}{ - } \left ( \color{#FF6800}{ 4 } \color{#FF6800}{ i } \right ) \end{array} $Get rid of unnecessary parentheses$ \begin{array} {l} x = 4 i \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ i } \end{array} $Do not have the solution$ $Calculate using the quadratic formula$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 48 } = 0 $Bind the expressions with the common factor$ 3\color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \right ) = 0\color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \right ) = \color{#FF6800}{ 0 } $Divide both sides by$ 3\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } = \color{#FF6800}{ 0 }x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 16 } } { 2 \times 1 } $0 has no sign$ x = \dfrac { \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times 16 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 16 } } { 2 \times 1 } $The power of 0 is 0$ x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times 16 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times 16 } } { 2 \times 1 } $0 does not change when you add or subtract$ x = \dfrac { 0 \pm \sqrt{ - 4 \times 1 \times 16 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times 16 } } { 2 \times 1 } $Multiplying any number by 1 does not change the value$ x = \dfrac { 0 \pm \sqrt{ - 4 \times 16 } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } } { 2 \times 1 } $Multiply$ - 4 $and$ 16x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 64 } } } { 2 \times 1 }x = \dfrac { 0 \pm \sqrt{ - 64 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } } $Multiplying any number by 1 does not change the value$ x = \dfrac { 0 \pm \sqrt{ - 64 } } { \color{#FF6800}{ 2 } }\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm \sqrt{ - 64 } } { 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}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 48 } = \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}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 48 }D = \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times 48 $The power of 0 is 0$ D = \color{#FF6800}{ 0 } - 4 \times 3 \times 48D = \color{#FF6800}{ 0 } - 4 \times 3 \times 48 $0 does not change when you add or subtract$ D = - 4 \times 3 \times 48D = \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 48 } $Multiply the numbers$ D = \color{#FF6800}{ - } \color{#FF6800}{ 576 }\color{#FF6800}{ D } = \color{#FF6800}{ - } \color{#FF6800}{ 576 } $Since$ D<0 $, there is no real root of the following quadratic equation$  $Do not have the real root$ \alpha + \beta = 0 , \alpha \beta = 16$Find the sum and product of the two roots of the quadratic equation$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 48 } = \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 } { 3 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 48 } { 3 } }\alpha + \beta = - \color{#FF6800}{ \dfrac { 0 } { 3 } } , \alpha \beta = \dfrac { 48 } { 3 } $If the numerator is 0, it is equal to 0$ \alpha + \beta = - \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { 48 } { 3 }\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { 48 } { 3 } $0 has no sign$ \alpha + \beta = \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { 48 } { 3 }\alpha + \beta = 0 , \alpha \beta = \color{#FF6800}{ \dfrac { 48 } { 3 } } $Reduce the fraction$ \alpha + \beta = 0 , \alpha \beta = \color{#FF6800}{ 16 } $그래프 보기$ \$
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