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How do you solve ${x^2} + 5x + 7 = 0$ using a quadratic formula?

If ${x^2} - 6x + 5 = 0$and ${x^2} - 12x + p = 0$ have a common root, then find the value of $p$.

Solve the following activity: zeroes of the polynomial ${{x}^{2}}-3x+2$ graphically.

If one root of the quadratic equation \[{{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\] is numerically equal but opposite in sign to one root of \[{{a}_{2}}{{x}^{2}}+{{b}_{2}}x+{{c}_{2}}=0\], then prove that the quadratic equation whose roots are the other roots of the both of these equations is $\dfrac{{{x}^{2}}}{\dfrac{{{b}_{1}}}{{{a}_{1}}}+\dfrac{{{b}_{2}}}{{{a}_{2}}}}+x+\dfrac{1}{\dfrac{{{b}_{1}}}{{{c}_{1}}}+\dfrac{{{b}_{2}}}{{{c}_{2}}}}=0$.

Find the quadratic equation whose sum and product of zeroes are $\dfrac{21}{8}$ and $\dfrac{5}{16}$ respectively.

Show that the equation ${{x}^{4}}-12{{x}^{2}}+12x-3=0$ has a root between -3 and -4 and another between 2 and 3.

If $\alpha ,\beta $ are roots of the equation ${{x}^{2}}-6x-2=0$ and we define ${{a}_{n}}={{\alpha }^{n}}-{{\beta }^{n}}$ then find the value of $\dfrac{{{a}_{10}}-2{{a}_{8}}}{2{{a}_{9}}}$

Consider the quadratic equation $(c-5){{x}^{2}}-2cx+(c-4)=0,c\ne 5$. Let S be the set of all integral values of c for which one root of the equation lies in the interval ( 0, 2 ) and it’s another root lies in the interval ( 2, 3 ). Then the number of elements in S is,

( a ) 11

( b ) 18

( c ) 10

( d ) 12

( a ) 11

( b ) 18

( c ) 10

( d ) 12

For which values of p is \[{{p}^{2}}-5p+6\] negative?

(a) p < 0

(b) 2 < p < 3

(c) p > 3

(d) p < 2

(a) p < 0

(b) 2 < p < 3

(c) p > 3

(d) p < 2

If the roots of \[a{x^2} + bx + c = 0\] are both negative and \[b < 0\] then

A) \[a < 0,c < 0\]

B) \[a < 0,c > 0\]

C) \[a > 0,c < 0\]

D) \[a > 0,c > 0\]

A) \[a < 0,c < 0\]

B) \[a < 0,c > 0\]

C) \[a > 0,c < 0\]

D) \[a > 0,c > 0\]

If one of the zeros of the quadratic polynomial of the form \[{{x}^{2}}+ax+b\] is negative of the other, then it

(a) has no linear term and the constant term is negative

(b) has no linear term and the constant term is positive

(c) can have a linear term but the constant term is negative

(d) can have a linear term but the constant term is positive.

(a) has no linear term and the constant term is negative

(b) has no linear term and the constant term is positive

(c) can have a linear term but the constant term is negative

(d) can have a linear term but the constant term is positive.

Find the zeroes of the polynomial $3{{x}^{2}}-2$ and verify the relationship between the zeroes and coefficients.

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