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Formula

Solve the quadratic equation

Answer

Number of solution

Answer

Relationship between roots and coefficients

Answer

Graph

$y = 0.3 x ^ { 2 } - 0.6 x$

$y = 1$

$x$Intercept

$\left ( 2 , 0 \right )$, $\left ( 0 , 0 \right )$

$y$Intercept

$\left ( 0 , 0 \right )$

Minimum

$\left ( 1 , - \dfrac { 3 } { 10 } \right )$

Standard form

$y = \dfrac { 3 } { 10 } \left ( x - 1 \right ) ^ { 2 } - \dfrac { 3 } { 10 }$

$0.3x ^{ 2 } -0.6x = 1$

$\begin{array} {l} x = \dfrac { 3 + \sqrt{ 39 } } { 3 } \\ x = \dfrac { 3 - \sqrt{ 39 } } { 3 } \end{array}$

Solve quadratic equations using the square root

$\color{#FF6800}{ 0.3 } x ^ { 2 } - 0.6 x = 1$

$ $ Convert decimals to fractions $ $

$\color{#FF6800}{ \dfrac { 3 } { 10 } } x ^ { 2 } - 0.6 x = 1$

$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 0.6 } x = 1$

$ $ Convert decimals to fractions $ $

$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 5 } } x = 1$

$\color{#FF6800}{ \dfrac { 3 } { 10 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 5 } } \color{#FF6800}{ x } = 1$

$ $ Calculate the expression as a fraction format $ $

$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 6 x } { 10 } } = 1$

$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 6 x } { 10 } } = \color{#FF6800}{ 1 }$

$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $

$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } = \color{#FF6800}{ 10 }$

$3 x ^ { 2 } - 6 x = \color{#FF6800}{ 10 }$

$ $ Move the expression to the left side and change the symbol $ $

$3 x ^ { 2 } - 6 x \color{#FF6800}{ - } \color{#FF6800}{ 10 } = 0$

$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 10 } = \color{#FF6800}{ 0 }$

$ $ Divide both sides by the coefficient of the leading highest term $ $

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 10 } { 3 } } = \color{#FF6800}{ 0 }$

$ $ Convert the quadratic expression on the left side to a perfect square format $ $

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 10 } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$

$ $ Organize the expression $ $

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 13 } { 3 } }$

$ $ Solve quadratic equations using the square root $ $

$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 13 } { 3 } } }$

$ $ Solve a solution to $ x$

$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 39 } } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 39 } } { 3 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 39 } } { 3 } } \end{array}$

$ $ Organize the expression $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 + \sqrt{ 39 } } { 3 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 - \sqrt{ 39 } } { 3 } } \end{array}$

$\begin{array} {l} x = \dfrac { 3 + \sqrt{ 39 } } { 3 } \\ x = \dfrac { 3 - \sqrt{ 39 } } { 3 } \end{array}$

Calculate using the quadratic formula

$\color{#FF6800}{ 0.3 } x ^ { 2 } - 0.6 x = 1$

$ $ Convert decimals to fractions $ $

$\color{#FF6800}{ \dfrac { 3 } { 10 } } x ^ { 2 } - 0.6 x = 1$

$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 0.6 } x = 1$

$ $ Convert decimals to fractions $ $

$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 5 } } x = 1$

$\color{#FF6800}{ \dfrac { 3 } { 10 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 5 } } \color{#FF6800}{ x } = 1$

$ $ Calculate the expression as a fraction format $ $

$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 6 x } { 10 } } = 1$

$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 6 x } { 10 } } = \color{#FF6800}{ 1 }$

$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $

$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } = \color{#FF6800}{ 10 }$

$3 x ^ { 2 } - 6 x = \color{#FF6800}{ 10 }$

$ $ Move the expression to the left side and change the symbol $ $

$3 x ^ { 2 } - 6 x \color{#FF6800}{ - } \color{#FF6800}{ 10 } = 0$

$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 6 \right ) \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 }$

$ $ Simplify Minus $ $

$x = \dfrac { 6 \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 }$

$x = \dfrac { 6 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 }$

$ $ Remove negative signs because negative numbers raised to even powers are positive $ $

$x = \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 } }$

$ $ Organize the expression $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 156 } } { 2 \times 3 } }$

$x = \dfrac { 6 \pm \sqrt{ \color{#FF6800}{ 156 } } } { 2 \times 3 }$

$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $

$x = \dfrac { 6 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 39 } } } { 2 \times 3 }$

$x = \dfrac { 6 \pm 2 \sqrt{ 39 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } }$

$ $ Multiply $ 2 $ and $ 3$

$x = \dfrac { 6 \pm 2 \sqrt{ 39 } } { \color{#FF6800}{ 6 } }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm 2 \sqrt{ 39 } } { 6 } }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 + 2 \sqrt{ 39 } } { 6 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 - 2 \sqrt{ 39 } } { 6 } } \end{array}$

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 6 + 2 \sqrt{ 39 } } { 6 } } \\ x = \dfrac { 6 - 2 \sqrt{ 39 } } { 6 } \end{array}$

$ $ Do the reduction of the fraction format $ $

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 3 + \sqrt{ 39 } } { 3 } } \\ x = \dfrac { 6 - 2 \sqrt{ 39 } } { 6 } \end{array}$

$\begin{array} {l} x = \dfrac { 3 + \sqrt{ 39 } } { 3 } \\ x = \color{#FF6800}{ \dfrac { 6 - 2 \sqrt{ 39 } } { 6 } } \end{array}$

$ $ Do the reduction of the fraction format $ $

$\begin{array} {l} x = \dfrac { 3 + \sqrt{ 39 } } { 3 } \\ x = \color{#FF6800}{ \dfrac { 3 - \sqrt{ 39 } } { 3 } } \end{array}$

$ $ 2 real roots $ $

Find the number of solutions

$\color{#FF6800}{ 0.3 } x ^ { 2 } - 0.6 x = 1$

$ $ Convert decimals to fractions $ $

$\color{#FF6800}{ \dfrac { 3 } { 10 } } x ^ { 2 } - 0.6 x = 1$

$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 0.6 } x = 1$

$ $ Convert decimals to fractions $ $

$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 5 } } x = 1$

$\color{#FF6800}{ \dfrac { 3 } { 10 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 5 } } \color{#FF6800}{ x } = 1$

$ $ Calculate the expression as a fraction format $ $

$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 6 x } { 10 } } = 1$

$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 6 x } { 10 } } = \color{#FF6800}{ 1 }$

$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $

$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } = \color{#FF6800}{ 10 }$

$3 x ^ { 2 } - 6 x = \color{#FF6800}{ 10 }$

$ $ Move the expression to the left side and change the symbol $ $

$3 x ^ { 2 } - 6 x \color{#FF6800}{ - } \color{#FF6800}{ 10 } = 0$

$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 10 } = \color{#FF6800}{ 0 }$

$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$

$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 10 } \right )$

$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times \left ( - 10 \right )$

$ $ Remove negative signs because negative numbers raised to even powers are positive $ $

$D = 6 ^ { 2 } - 4 \times 3 \times \left ( - 10 \right )$

$D = \color{#FF6800}{ 6 } ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times \left ( - 10 \right )$

$ $ Calculate power $ $

$D = \color{#FF6800}{ 36 } - 4 \times 3 \times \left ( - 10 \right )$

$D = 36 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 10 } \right )$

$ $ Multiply the numbers $ $

$D = 36 + \color{#FF6800}{ 120 }$

$D = \color{#FF6800}{ 36 } \color{#FF6800}{ + } \color{#FF6800}{ 120 }$

$ $ Add $ 36 $ and $ 120$

$D = \color{#FF6800}{ 156 }$

$\color{#FF6800}{ D } = \color{#FF6800}{ 156 }$

$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $

$ $ 2 real roots $ $

$\alpha + \beta = 2 , \alpha \beta = - \dfrac { 10 } { 3 }$

Find the sum and product of the two roots of the quadratic equation

$\color{#FF6800}{ 0.3 } x ^ { 2 } - 0.6 x = 1$

$ $ Convert decimals to fractions $ $

$\color{#FF6800}{ \dfrac { 3 } { 10 } } x ^ { 2 } - 0.6 x = 1$

$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 0.6 } x = 1$

$ $ Convert decimals to fractions $ $

$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 5 } } x = 1$

$\color{#FF6800}{ \dfrac { 3 } { 10 } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 5 } } \color{#FF6800}{ x } = 1$

$ $ Calculate the expression as a fraction format $ $

$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 6 x } { 10 } } = 1$

$\color{#FF6800}{ \dfrac { 3 x ^ { 2 } - 6 x } { 10 } } = \color{#FF6800}{ 1 }$

$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $

$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } = \color{#FF6800}{ 10 }$

$3 x ^ { 2 } - 6 x = \color{#FF6800}{ 10 }$

$ $ Move the expression to the left side and change the symbol $ $

$3 x ^ { 2 } - 6 x \color{#FF6800}{ - } \color{#FF6800}{ 10 } = 0$

$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 10 } = \color{#FF6800}{ 0 }$

$ $ 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}$

$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 6 } { 3 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 10 } { 3 } }$

$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 6 } { 3 } } , \alpha \beta = \dfrac { - 10 } { 3 }$

$ $ Solve the sign of a fraction with a negative sign $ $

$\alpha + \beta = \color{#FF6800}{ \dfrac { 6 } { 3 } } , \alpha \beta = \dfrac { - 10 } { 3 }$

$ $ Reduce the fraction $ $

$\alpha + \beta = \color{#FF6800}{ 2 } , \alpha \beta = \dfrac { - 10 } { 3 }$

$\alpha + \beta = 2 , \alpha \beta = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 10 } } { 3 }$

$ $ Move the minus sign to the front of the fraction $ $

$\alpha + \beta = 2 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 10 } { 3 } }$

Solution search results

$11\right)\dfrac {m} {4}=-1$ $13\right)$ $20=15+y$ $15\right)$ $-0.6x=11.22$ $17\right)\dfrac {n} {8.2}=-3.7$

7th-9th grade

Algebra

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