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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
$y = 15 x ^ { 2 } + 14 x - 15$
$y = 0$
$x$Intercept
$\left ( - \dfrac { 7 } { 15 } + \dfrac { \sqrt{ 274 } } { 15 } , 0 \right )$, $\left ( - \dfrac { \sqrt{ 274 } } { 15 } - \dfrac { 7 } { 15 } , 0 \right )$
$y$Intercept
$\left ( 0 , - 15 \right )$
Minimum
$\left ( - \dfrac { 7 } { 15 } , - \dfrac { 274 } { 15 } \right )$
Standard form
$y = 15 \left ( x + \dfrac { 7 } { 15 } \right ) ^ { 2 } - \dfrac { 274 } { 15 }$
$15x ^{ 2 } +14x-15 = 0$
$\begin{array} {l} x = \dfrac { - 7 + \sqrt{ 274 } } { 15 } \\ x = \dfrac { - 7 - \sqrt{ 274 } } { 15 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 15 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 14 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 15 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 14 } { 15 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 14 } { 15 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 7 } { 15 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 7 } { 15 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x + \dfrac { 7 } { 15 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 7 } { 15 } } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x + \dfrac { 7 } { 15 } \right ) ^ { 2 } = \color{#FF6800}{ 1 } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { 7 } { 15 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$\left ( x + \dfrac { 7 } { 15 } \right ) ^ { 2 } = 1 + \left ( \color{#FF6800}{ \dfrac { 7 } { 15 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$ $ When raising a fraction to the power, raise the numerator and denominator each to the power $ $
$\left ( x + \dfrac { 7 } { 15 } \right ) ^ { 2 } = 1 + \dfrac { \color{#FF6800}{ 7 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 15 } ^ { \color{#FF6800}{ 2 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 7 } { 15 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 7 ^ { 2 } } { 15 ^ { 2 } } }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 7 } { 15 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 274 } { 225 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 7 } { 15 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 274 } { 225 } }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 7 } { 15 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 274 } { 225 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 7 } { 15 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 274 } { 225 } } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 274 } } { 15 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 7 } { 15 } }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 274 } } { 15 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 7 } { 15 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 7 } { 15 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 274 } } { 15 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 7 } { 15 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 274 } } { 15 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 7 } { 15 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 274 } } { 15 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 7 } { 15 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 274 } } { 15 } } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 7 + \sqrt{ 274 } } { 15 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 7 - \sqrt{ 274 } } { 15 } } \end{array}$
$\begin{array} {l} x = \dfrac { - 7 + \sqrt{ 274 } } { 15 } \\ x = \dfrac { - 7 - \sqrt{ 274 } } { 15 } \end{array}$
Calculate using the quadratic formula
$x = \dfrac { - 14 \pm \sqrt{ \color{#FF6800}{ 14 } ^ { \color{#FF6800}{ 2 } } - 4 \times 15 \times \left ( - 15 \right ) } } { 2 \times 15 }$
$ $ Calculate power $ $
$x = \dfrac { - 14 \pm \sqrt{ \color{#FF6800}{ 196 } - 4 \times 15 \times \left ( - 15 \right ) } } { 2 \times 15 }$
$x = \dfrac { - 14 \pm \sqrt{ 196 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 15 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 15 } \right ) } } { 2 \times 15 }$
$ $ Multiply the numbers $ $
$x = \dfrac { - 14 \pm \sqrt{ 196 + \color{#FF6800}{ 900 } } } { 2 \times 15 }$
$x = \dfrac { - 14 \pm \sqrt{ \color{#FF6800}{ 196 } \color{#FF6800}{ + } \color{#FF6800}{ 900 } } } { 2 \times 15 }$
$ $ Add $ 196 $ and $ 900$
$x = \dfrac { - 14 \pm \sqrt{ \color{#FF6800}{ 1096 } } } { 2 \times 15 }$
$x = \dfrac { - 14 \pm \sqrt{ \color{#FF6800}{ 1096 } } } { 2 \times 15 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 14 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 274 } } } { 2 \times 15 }$
$x = \dfrac { - 14 \pm 2 \sqrt{ 274 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 15 } }$
$ $ Multiply $ 2 $ and $ 15$
$x = \dfrac { - 14 \pm 2 \sqrt{ 274 } } { \color{#FF6800}{ 30 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 14 \pm 2 \sqrt{ 274 } } { 30 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 14 + 2 \sqrt{ 274 } } { 30 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 14 - 2 \sqrt{ 274 } } { 30 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 14 + 2 \sqrt{ 274 } } { 30 } } \\ x = \dfrac { - 14 - 2 \sqrt{ 274 } } { 30 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 7 + \sqrt{ 274 } } { 15 } } \\ x = \dfrac { - 14 - 2 \sqrt{ 274 } } { 30 } \end{array}$
$\begin{array} {l} x = \dfrac { - 7 + \sqrt{ 274 } } { 15 } \\ x = \color{#FF6800}{ \dfrac { - 14 - 2 \sqrt{ 274 } } { 30 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \dfrac { - 7 + \sqrt{ 274 } } { 15 } \\ x = \color{#FF6800}{ \dfrac { - 7 - \sqrt{ 274 } } { 15 } } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\color{#FF6800}{ 15 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 14 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 15 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 14 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 15 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 15 } \right )$
$D = \color{#FF6800}{ 14 } ^ { \color{#FF6800}{ 2 } } - 4 \times 15 \times \left ( - 15 \right )$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 196 } - 4 \times 15 \times \left ( - 15 \right )$
$D = 196 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 15 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 15 } \right )$
$ $ Multiply the numbers $ $
$D = 196 + \color{#FF6800}{ 900 }$
$D = \color{#FF6800}{ 196 } \color{#FF6800}{ + } \color{#FF6800}{ 900 }$
$ $ Add $ 196 $ and $ 900$
$D = \color{#FF6800}{ 1096 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 1096 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = - \dfrac { 14 } { 15 } , \alpha \beta = - 1$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 15 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 14 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 15 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 14 } { 15 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 15 } { 15 } }$
$\alpha + \beta = - \dfrac { 14 } { 15 } , \alpha \beta = \color{#FF6800}{ \dfrac { - 15 } { 15 } }$
$ $ Reduce the fraction $ $
$\alpha + \beta = - \dfrac { 14 } { 15 } , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
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