Symbol

# Calculator search results

Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = 4 x ^ { 2 } - 24 x + 36$
$y = 0$
$x$Intercept
$\left ( 3 , 0 \right )$
$y$Intercept
$\left ( 0 , 36 \right )$
Minimum
$\left ( 3 , 0 \right )$
Standard form
$y = 4 \left ( x - 3 \right ) ^ { 2 }$
$4x ^{ 2 } -24x+36 = 0$
$x = 3$
Solve quadratic equations using the square root
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 24 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 36 } = \color{#FF6800}{ 0 }$
 Divide both sides by the coefficient of the leading highest term 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = 0$
 Express as the perfect square formula 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \color{#FF6800}{ 3 }$
$x = 3$
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 24 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 36 } = 0$
 Bind the expressions with the common factor $4$
$\color{#FF6800}{ 4 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } \right ) = 0$
$\color{#FF6800}{ 4 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } \right ) = \color{#FF6800}{ 0 }$
 Divide both sides by $4$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 6 \right ) \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$
 Simplify Minus 
$x = \dfrac { 6 \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$
$x = \dfrac { 6 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 9 } } { 2 \times 1 }$
 Remove negative signs because negative numbers raised to even powers are positive 
$x = \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 } }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 0 } } { 2 \times 1 } }$
$x = \dfrac { 6 \pm \sqrt{ \color{#FF6800}{ 0 } } } { 2 \times 1 }$
$n square root$ of 0 is 0 
$x = \dfrac { 6 \pm \color{#FF6800}{ 0 } } { 2 \times 1 }$
$x = \dfrac { 6 \pm 0 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
 Multiplying any number by 1 does not change the value 
$x = \dfrac { 6 \pm 0 } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm 0 } { 2 } }$
 The value will not be changed even if adding or subtracting 0 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 } { 2 } }$
$x = \color{#FF6800}{ \dfrac { 6 } { 2 } }$
 Do the reduction of the fraction format 
$x = \color{#FF6800}{ \dfrac { 3 } { 1 } }$
$x = \color{#FF6800}{ \dfrac { 3 } { 1 } }$
 Reduce the fraction to the lowest term 
$x = \color{#FF6800}{ 3 }$
 1 real root (multiple root) 
Find the number of solutions
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 24 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 36 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 24 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 36 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 24 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times 36$
 Remove negative signs because negative numbers raised to even powers are positive 
$D = 24 ^ { 2 } - 4 \times 4 \times 36$
$D = \color{#FF6800}{ 24 } ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times 36$
 Calculate power 
$D = \color{#FF6800}{ 576 } - 4 \times 4 \times 36$
$D = 576 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 36 }$
 Multiply the numbers 
$D = 576 \color{#FF6800}{ - } \color{#FF6800}{ 576 }$
$D = \color{#FF6800}{ 576 } \color{#FF6800}{ - } \color{#FF6800}{ 576 }$
 Remove the two numbers if the values are the same and the signs are different 
$D = 0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 0 }$
 Since $D=0$ , the number of real root of the following quadratic equation is 1 (multiple root) 
 1 real root (multiple root) 
$\alpha + \beta = 6 , \alpha \beta = 9$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 24 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 36 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 24 } { 4 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 36 } { 4 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 24 } { 4 } } , \alpha \beta = \dfrac { 36 } { 4 }$
 Solve the sign of a fraction with a negative sign 
$\alpha + \beta = \color{#FF6800}{ \dfrac { 24 } { 4 } } , \alpha \beta = \dfrac { 36 } { 4 }$
$\alpha + \beta = \color{#FF6800}{ \dfrac { 24 } { 4 } } , \alpha \beta = \dfrac { 36 } { 4 }$
 Reduce the fraction 
$\alpha + \beta = \color{#FF6800}{ 6 } , \alpha \beta = \dfrac { 36 } { 4 }$
$\alpha + \beta = 6 , \alpha \beta = \color{#FF6800}{ \dfrac { 36 } { 4 } }$
 Reduce the fraction 
$\alpha + \beta = 6 , \alpha \beta = \color{#FF6800}{ 9 }$
Solution search results
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