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Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = 0.5 x ^ { 2 } - \dfrac { 2 } { 3 } x - \dfrac { 5 } { 6 }$
$y = 0$
$x$Intercept
$\left ( \dfrac { 2 } { 3 } + \dfrac { \sqrt{ 19 } } { 3 } , 0 \right )$, $\left ( \dfrac { 2 } { 3 } - \dfrac { \sqrt{ 19 } } { 3 } , 0 \right )$
$y$Intercept
$\left ( 0 , - \dfrac { 5 } { 6 } \right )$
Minimum
$\left ( \dfrac { 2 } { 3 } , - \dfrac { 19 } { 18 } \right )$
Standard form
$y = \dfrac { 1 } { 2 } \left ( x - \dfrac { 2 } { 3 } \right ) ^ { 2 } - \dfrac { 19 } { 18 }$
$0.5x ^{ 2 } - \dfrac{ 2 }{ 3 } x- \dfrac{ 5 }{ 6 } = 0$
$\begin{array} {l} x = \dfrac { 2 + \sqrt{ 19 } } { 3 } \\ x = \dfrac { 2 - \sqrt{ 19 } } { 3 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 0.5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 6 } } = 0$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 4 x - 5 } { 6 } } = 0$
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 4 x - 5 } { 6 } } = \color{#FF6800}{ 0 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 }$
 Divide both sides by the coefficient of the leading highest term 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 4 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 3 } } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 4 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 3 } } = \color{#FF6800}{ 0 }$
 Convert the quadratic expression on the left side to a perfect square format 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 3 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 2 } { 3 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x - \dfrac { 2 } { 3 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 3 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 2 } { 3 } } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
 Move the constant to the right side and change the sign 
$\left ( x - \dfrac { 2 } { 3 } \right ) ^ { 2 } = \color{#FF6800}{ \dfrac { 5 } { 3 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { 2 } { 3 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$\left ( x - \dfrac { 2 } { 3 } \right ) ^ { 2 } = \dfrac { 5 } { 3 } + \left ( \color{#FF6800}{ \dfrac { 2 } { 3 } } \right ) ^ { \color{#FF6800}{ 2 } }$
 When raising a fraction to the power, raise the numerator and denominator each to the power 
$\left ( x - \dfrac { 2 } { 3 } \right ) ^ { 2 } = \dfrac { 5 } { 3 } + \dfrac { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 5 } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 2 ^ { 2 } } { 3 ^ { 2 } } }$
 Organize the expression 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 19 } { 9 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 19 } { 9 } }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 19 } { 9 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 19 } { 9 } } }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 19 } } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 2 } { 3 } }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 19 } } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 2 } { 3 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 19 } } { 3 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 19 } } { 3 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 19 } } { 3 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 19 } } { 3 } } \end{array}$
 Organize the expression 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 2 + \sqrt{ 19 } } { 3 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 2 - \sqrt{ 19 } } { 3 } } \end{array}$
$\begin{array} {l} x = \dfrac { 2 + \sqrt{ 19 } } { 3 } \\ x = \dfrac { 2 - \sqrt{ 19 } } { 3 } \end{array}$
$\color{#FF6800}{ 0.5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 6 } } = 0$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 4 x - 5 } { 6 } } = 0$
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 4 x - 5 } { 6 } } = \color{#FF6800}{ 0 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 4 \right ) \pm \sqrt{ \left ( - 4 \right ) ^ { 2 } - 4 \times 3 \times \left ( - 5 \right ) } } { 2 \times 3 }$
 Simplify Minus 
$x = \dfrac { 4 \pm \sqrt{ \left ( - 4 \right ) ^ { 2 } - 4 \times 3 \times \left ( - 5 \right ) } } { 2 \times 3 }$
$x = \dfrac { 4 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times \left ( - 5 \right ) } } { 2 \times 3 }$
 Remove negative signs because negative numbers raised to even powers are positive 
$x = \dfrac { 4 \pm \sqrt{ 4 ^ { 2 } - 4 \times 3 \times \left ( - 5 \right ) } } { 2 \times 3 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 \pm \sqrt{ 4 ^ { 2 } - 4 \times 3 \times \left ( - 5 \right ) } } { 2 \times 3 } }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 \pm \sqrt{ 76 } } { 2 \times 3 } }$
$x = \dfrac { 4 \pm \sqrt{ \color{#FF6800}{ 76 } } } { 2 \times 3 }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$x = \dfrac { 4 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 19 } } } { 2 \times 3 }$
$x = \dfrac { 4 \pm 2 \sqrt{ 19 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } }$
 Multiply $2$ and $3$
$x = \dfrac { 4 \pm 2 \sqrt{ 19 } } { \color{#FF6800}{ 6 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 \pm 2 \sqrt{ 19 } } { 6 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 + 2 \sqrt{ 19 } } { 6 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 - 2 \sqrt{ 19 } } { 6 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 4 + 2 \sqrt{ 19 } } { 6 } } \\ x = \dfrac { 4 - 2 \sqrt{ 19 } } { 6 } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 2 + \sqrt{ 19 } } { 3 } } \\ x = \dfrac { 4 - 2 \sqrt{ 19 } } { 6 } \end{array}$
$\begin{array} {l} x = \dfrac { 2 + \sqrt{ 19 } } { 3 } \\ x = \color{#FF6800}{ \dfrac { 4 - 2 \sqrt{ 19 } } { 6 } } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = \dfrac { 2 + \sqrt{ 19 } } { 3 } \\ x = \color{#FF6800}{ \dfrac { 2 - \sqrt{ 19 } } { 3 } } \end{array}$
 2 real roots 
Find the number of solutions
$\color{#FF6800}{ 0.5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 6 } } = 0$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 4 x - 5 } { 6 } } = 0$
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 4 x - 5 } { 6 } } = \color{#FF6800}{ 0 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right )$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times \left ( - 5 \right )$
 Remove negative signs because negative numbers raised to even powers are positive 
$D = 4 ^ { 2 } - 4 \times 3 \times \left ( - 5 \right )$
$D = \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times \left ( - 5 \right )$
 Calculate power 
$D = \color{#FF6800}{ 16 } - 4 \times 3 \times \left ( - 5 \right )$
$D = 16 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right )$
 Multiply the numbers 
$D = 16 + \color{#FF6800}{ 60 }$
$D = \color{#FF6800}{ 16 } \color{#FF6800}{ + } \color{#FF6800}{ 60 }$
 Add $16$ and $60$
$D = \color{#FF6800}{ 76 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 76 }$
 Since $D>0$ , the number of real root of the following quadratic equation is 2 
 2 real roots 
$\alpha + \beta = \dfrac { 4 } { 3 } , \alpha \beta = - \dfrac { 5 } { 3 }$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 0.5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 6 } } = 0$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 4 x - 5 } { 6 } } = 0$
$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 4 x - 5 } { 6 } } = \color{#FF6800}{ 0 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 4 } { 3 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 5 } { 3 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 4 } { 3 } } , \alpha \beta = \dfrac { - 5 } { 3 }$
 Solve the sign of a fraction with a negative sign 
$\alpha + \beta = \color{#FF6800}{ \dfrac { 4 } { 3 } } , \alpha \beta = \dfrac { - 5 } { 3 }$
$\alpha + \beta = \dfrac { 4 } { 3 } , \alpha \beta = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 5 } } { 3 }$
 Move the minus sign to the front of the fraction 
$\alpha + \beta = \dfrac { 4 } { 3 } , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 3 } }$
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