qanda-logo
apple logogoogle play logo

Calculator search results

Formula
Solve the quadratic equation
Answer
circle-check-icon
expand-arrow-icon
expand-arrow-icon
expand-arrow-icon
expand-arrow-icon
expand-arrow-icon
expand-arrow-icon
Number of solution
Answer
circle-check-icon
expand-arrow-icon
expand-arrow-icon
expand-arrow-icon
Relationship between roots and coefficients
Answer
circle-check-icon
expand-arrow-icon
expand-arrow-icon
Graph
$y = 0.5 x ^ { 2 } + \dfrac { 1 } { 6 } x - \dfrac { 1 } { 3 }$
$y = 0$
$x$Intercept
$\left ( - 1 , 0 \right )$, $\left ( \dfrac { 2 } { 3 } , 0 \right )$
$y$Intercept
$\left ( 0 , - \dfrac { 1 } { 3 } \right )$
Minimum
$\left ( - \dfrac { 1 } { 6 } , - \dfrac { 25 } { 72 } \right )$
Standard form
$y = \dfrac { 1 } { 2 } \left ( x + \dfrac { 1 } { 6 } \right ) ^ { 2 } - \dfrac { 25 } { 72 }$
$0.5x ^{ 2 } + \dfrac{ 1 }{ 6 } x- \dfrac{ 1 }{ 3 } = 0$
$\begin{array} {l} x = \dfrac { 2 } { 3 } \\ x = - 1 \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 0.5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } = 0$
$ $ Calculate the expression as a fraction format $ $
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } + x - 2 } { 6 } } = 0$
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } + x - 2 } { 6 } } = \color{#FF6800}{ 0 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 1 } { 6 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x + \dfrac { 1 } { 6 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 1 } { 6 } } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x + \dfrac { 1 } { 6 } \right ) ^ { 2 } = \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { 1 } { 6 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$\left ( x + \dfrac { 1 } { 6 } \right ) ^ { 2 } = \dfrac { 2 } { 3 } + \left ( \color{#FF6800}{ \dfrac { 1 } { 6 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$ $ When raising a fraction to the power, raise the numerator and denominator each to the power $ $
$\left ( x + \dfrac { 1 } { 6 } \right ) ^ { 2 } = \dfrac { 2 } { 3 } + \dfrac { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 6 } ^ { \color{#FF6800}{ 2 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 ^ { 2 } } { 6 ^ { 2 } } }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 25 } { 36 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 25 } { 36 } }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 25 } { 36 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 25 } { 36 } } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 5 } { 6 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 6 } }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 5 } { 6 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 6 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 6 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 5 } { 6 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 6 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 6 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 6 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 5 } { 6 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 6 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 6 } } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 2 } { 3 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \end{array}$
$\begin{array} {l} x = \dfrac { 2 } { 3 } \\ x = - 1 \end{array}$
Calculate using the quadratic formula
$\color{#FF6800}{ 0.5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } = 0$
$ $ Calculate the expression as a fraction format $ $
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } + x - 2 } { 6 } } = 0$
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } + x - 2 } { 6 } } = \color{#FF6800}{ 0 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm \sqrt{ 1 ^ { 2 } - 4 \times 3 \times \left ( - 2 \right ) } } { 2 \times 3 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm \sqrt{ 25 } } { 2 \times 3 } }$
$x = \dfrac { - 1 \pm \sqrt{ \color{#FF6800}{ 25 } } } { 2 \times 3 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 1 \pm \color{#FF6800}{ 5 } } { 2 \times 3 }$
$x = \dfrac { - 1 \pm 5 } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } }$
$ $ Multiply $ 2 $ and $ 3$
$x = \dfrac { - 1 \pm 5 } { \color{#FF6800}{ 6 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm 5 } { 6 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 + 5 } { 6 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 - 5 } { 6 } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 5 } } { 6 } \\ x = \dfrac { - 1 - 5 } { 6 } \end{array}$
$ $ Add $ - 1 $ and $ 5$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 4 } } { 6 } \\ x = \dfrac { - 1 - 5 } { 6 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 4 } { 6 } } \\ x = \dfrac { - 1 - 5 } { 6 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 2 } { 3 } } \\ x = \dfrac { - 1 - 5 } { 6 } \end{array}$
$\begin{array} {l} x = \dfrac { 2 } { 3 } \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } } { 6 } \end{array}$
$ $ Find the sum of the negative numbers $ $
$\begin{array} {l} x = \dfrac { 2 } { 3 } \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 6 } } { 6 } \end{array}$
$\begin{array} {l} x = \dfrac { 2 } { 3 } \\ x = \color{#FF6800}{ \dfrac { - 6 } { 6 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \dfrac { 2 } { 3 } \\ x = \color{#FF6800}{ \dfrac { - 1 } { 1 } } \end{array}$
$\begin{array} {l} x = \dfrac { 2 } { 3 } \\ x = \dfrac { - 1 } { \color{#FF6800}{ 1 } } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = \dfrac { 2 } { 3 } \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\color{#FF6800}{ 0.5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } = 0$
$ $ Calculate the expression as a fraction format $ $
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } + x - 2 } { 6 } } = 0$
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } + x - 2 } { 6 } } = \color{#FF6800}{ 0 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right )$
$D = \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times \left ( - 2 \right )$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 1 } - 4 \times 3 \times \left ( - 2 \right )$
$D = 1 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right )$
$ $ Multiply the numbers $ $
$D = 1 + \color{#FF6800}{ 24 }$
$D = \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 24 }$
$ $ Add $ 1 $ and $ 24$
$D = \color{#FF6800}{ 25 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 25 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = - \dfrac { 1 } { 3 } , \alpha \beta = - \dfrac { 2 } { 3 }$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 0.5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 6 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } = 0$
$ $ Calculate the expression as a fraction format $ $
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } + x - 2 } { 6 } } = 0$
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } + x - 2 } { 6 } } = \color{#FF6800}{ 0 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 2 } { 3 } }$
$\alpha + \beta = - \dfrac { 1 } { 3 } , \alpha \beta = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 2 } } { 3 }$
$ $ Move the minus sign to the front of the fraction $ $
$\alpha + \beta = - \dfrac { 1 } { 3 } , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } }$
$ $ 그래프 보기 $ $
Graph
Solution search results
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
Have you found the solution you wanted?
Try again
Try more features at Qanda!
Search by problem image
Ask 1:1 question to TOP class teachers
AI recommend problems and video lecture
apple logogoogle play logo