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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
$y = 2 x ^ { 2 } - x - 21$
$y = 0$
$x$Intercept
$\left ( \dfrac { 7 } { 2 } , 0 \right )$, $\left ( - 3 , 0 \right )$
$y$Intercept
$\left ( 0 , - 21 \right )$
Minimum
$\left ( \dfrac { 1 } { 4 } , - \dfrac { 169 } { 8 } \right )$
Standard form
$y = 2 \left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } - \dfrac { 169 } { 8 }$
$2x ^{ 2 } -x-21 = 0$
$\begin{array} {l} x = - 3 \\ x = \dfrac { 7 } { 2 } \end{array}$
Find solution by method of factorization
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 21 } = 0$
$acx^{2} + \left(ad + bc\right)x +bd = \left(ax + b\right)\left(cx+d\right)$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) = 0$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) = \color{#FF6800}{ 0 }$
$ $ If the product of the factor is 0, at least one factor should be 0 $ $
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 } \end{array}$
$ $ Solve the equation to find $ x$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 7 } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { 7 } { 2 } \\ x = - 3 \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 21 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 21 } { 2 } } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 21 } { 2 } } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 4 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 21 } { 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 1 } { 4 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 21 } { 2 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 1 } { 4 } } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } = \color{#FF6800}{ \dfrac { 21 } { 2 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { 1 } { 4 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$\left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } = \dfrac { 21 } { 2 } + \left ( \color{#FF6800}{ \dfrac { 1 } { 4 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$ $ When raising a fraction to the power, raise the numerator and denominator each to the power $ $
$\left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } = \dfrac { 21 } { 2 } + \dfrac { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 4 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 21 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 ^ { 2 } } { 4 ^ { 2 } } }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 4 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 169 } { 16 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 4 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 169 } { 16 } }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 4 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 169 } { 16 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 4 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 169 } { 16 } } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 13 } { 4 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 4 } }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 13 } { 4 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 4 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 } { 4 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 13 } { 4 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 } { 4 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 13 } { 4 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 } { 4 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 13 } { 4 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 } { 4 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 13 } { 4 } } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 7 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \end{array}$
$\begin{array} {l} x = \dfrac { 7 } { 2 } \\ x = - 3 \end{array}$
Calculate using the quadratic formula
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 1 \right ) \pm \sqrt{ \left ( - 1 \right ) ^ { 2 } - 4 \times 2 \times \left ( - 21 \right ) } } { 2 \times 2 }$
$ $ Simplify Minus $ $
$x = \dfrac { 1 \pm \sqrt{ \left ( - 1 \right ) ^ { 2 } - 4 \times 2 \times \left ( - 21 \right ) } } { 2 \times 2 }$
$x = \dfrac { 1 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 21 \right ) } } { 2 \times 2 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 1 \pm \sqrt{ 1 ^ { 2 } - 4 \times 2 \times \left ( - 21 \right ) } } { 2 \times 2 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 \pm \sqrt{ 1 ^ { 2 } - 4 \times 2 \times \left ( - 21 \right ) } } { 2 \times 2 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 \pm \sqrt{ 169 } } { 2 \times 2 } }$
$x = \dfrac { 1 \pm \sqrt{ \color{#FF6800}{ 169 } } } { 2 \times 2 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 1 \pm \color{#FF6800}{ 13 } } { 2 \times 2 }$
$x = \dfrac { 1 \pm 13 } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } }$
$ $ Multiply $ 2 $ and $ 2$
$x = \dfrac { 1 \pm 13 } { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 \pm 13 } { 4 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 + 13 } { 4 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 1 - 13 } { 4 } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 13 } } { 4 } \\ x = \dfrac { 1 - 13 } { 4 } \end{array}$
$ $ Add $ 1 $ and $ 13$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 14 } } { 4 } \\ x = \dfrac { 1 - 13 } { 4 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 14 } { 4 } } \\ x = \dfrac { 1 - 13 } { 4 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 7 } { 2 } } \\ x = \dfrac { 1 - 13 } { 4 } \end{array}$
$\begin{array} {l} x = \dfrac { 7 } { 2 } \\ x = \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 13 } } { 4 } \end{array}$
$ $ Subtract $ 13 $ from $ 1$
$\begin{array} {l} x = \dfrac { 7 } { 2 } \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 12 } } { 4 } \end{array}$
$\begin{array} {l} x = \dfrac { 7 } { 2 } \\ x = \color{#FF6800}{ \dfrac { - 12 } { 4 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \dfrac { 7 } { 2 } \\ x = \color{#FF6800}{ \dfrac { - 3 } { 1 } } \end{array}$
$\begin{array} {l} x = \dfrac { 7 } { 2 } \\ x = \dfrac { - 3 } { \color{#FF6800}{ 1 } } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = \dfrac { 7 } { 2 } \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 21 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 21 } \right )$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 21 \right )$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$D = 1 ^ { 2 } - 4 \times 2 \times \left ( - 21 \right )$
$D = \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 21 \right )$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 1 } - 4 \times 2 \times \left ( - 21 \right )$
$D = 1 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 21 } \right )$
$ $ Multiply the numbers $ $
$D = 1 + \color{#FF6800}{ 168 }$
$D = \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 168 }$
$ $ Add $ 1 $ and $ 168$
$D = \color{#FF6800}{ 169 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 169 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = \dfrac { 1 } { 2 } , \alpha \beta = - \dfrac { 21 } { 2 }$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 21 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 1 } { 2 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 21 } { 2 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 1 } { 2 } } , \alpha \beta = \dfrac { - 21 } { 2 }$
$ $ Solve the sign of a fraction with a negative sign $ $
$\alpha + \beta = \color{#FF6800}{ \dfrac { 1 } { 2 } } , \alpha \beta = \dfrac { - 21 } { 2 }$
$\alpha + \beta = \dfrac { 1 } { 2 } , \alpha \beta = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 21 } } { 2 }$
$ $ Move the minus sign to the front of the fraction $ $
$\alpha + \beta = \dfrac { 1 } { 2 } , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 21 } { 2 } }$
$ $ 그래프 보기 $ $
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Solution search results
search-thumbnail-PRACTICE QUESTIONS 
CLASS X : CHAPTER - 4 
$vADRAx1C$ EQUATIONS 
FACTORISATION METHOD 
Solve the following quadratic equations: 
1. $x^{2}+11x+30=0$ $2430x^{2}+7x-15=0$ 
+ 
2. $x^{2}+18x+32=0$ $25.24x^{2}-41x+12=0$ 
- 
3. $x^{2}+7x-18=0$ $26.2x^{2}-7x-15=0$ 
- - 
4. x² + 5x 6 - 0 $27.6x^{2}+11x-10=0$ 
– - 
5. y- 4y + 3 =0 $28.10x^{2}-9x-7=0$ 
- 
6. $x^{2}-21x+108=0$ $29.5x^{2}-16x-21=0$ 
- – 
7. $x^{2}-11x-80=0$ $30.2x^{2}-x-21=0$ 
8. x-x- 156 -0 $31.15x^{2}-x-28=0$ 
- 
9. $x^{2}-32x-105=0$ $32.8x^{2}-27b+9b^{2}=0$ 
- 
$1040+3x-x^{2}=0$ $33.5x^{2}+33xy-14y^{2}=0$ 
– –
10th-13th grade
Algebra
search-thumbnail-1. x + 11x + 30 = 0 $24.30x^{2}+7x-15=0$ 
2. x + 18x + 32 = 0 $25.24x^{2}-41x+12=0$ 
3. x + 7x – 18 = 0 $26.2x^{2}-7x-15=0$ 
4. x? + 5x – 6 = 0 $27.6x^{2}+11x-10=0$ 
5. y - 4y + 3 = 0 $28.10x^{2}-9x-7=0$ 
6. $x^{2}-21x+108=0$ $29.5x^{2}-16x-21=0$ 
7. x - 11x – 80 = 0 $30.2x^{2}-x-21=0$ 
8. x? - x- 156 = 0 $31.15x^{2}-x-28=0$ 
9. 2- 32z – 105 = 0 $32.8a^{2}-27ab+9b^{2}$ = 0 
10.40 + 3x – x = 0 $33.5x^{2}+33xy-14y^{2}$ = 0 
11.6 – x - x = 0 $34.3x^{3}-x^{2}-10x=0$ 
$12.7x^{2}+49x+84$ = 0 $35.x^{2}+9x+18=0$ 
$13.m^{2}+17mn-84n^{2}$ =0 $36.x^{2}+5x-24=0$ 
$14.5x^{2}+16x+3=0$ $37.x^{2}-4x-21=0$
10th-13th grade
Algebra
search-thumbnail-$∠-114=1$ 
$7\right)$ $Directi0n:$ Solve for the value of the discriminant and characterize the 
nature of zeros in each quadratic function. 
$ax^{2}+bx+c=0$ Discriminant Nature of Roots 
$1$ $x^{2}-2x+4=0$ 
$2$ $x^{2}+4x-21=0$ 
$3.$ $x^{2}+3x+3=0$ 
$4$ $x^{2}-5x+12=0$ 
$5$ $x^{2}-9x+7=0$ 
$6$ $-2x^{2}-x-1=0$ 
$7.$ $x^{2}+5x+2=0$
7th-9th grade
Other
search-thumbnail-ACTIVITY $2.$ $2-1N-1$ 
$Direction:$ Solve for the value of the discriminant and characterize the 
nature of zeros in each quadratic function. 
$ax^{2}+bx+c=0$ Discriminant Nature of Roots 
$1$ $x^{2}-2x-4=0$ 
$2$ $x^{2}+4x-21=0$ 
$3$ $x^{2}+3x+3=0$ 
$4$ $x^{2}-5x+12=0$ 
$5.$ $x^{2}-9x+7=0$ 
$-2x^{2}-x-1=0$ 
$6.$ 
$x^{2}+5x+2=0$ 
$7.$
7th-9th grade
Other
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