Question

the roots of a quadratic equation are 5 and -2. find the equation

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Answer 1

$\huge{\bf{\green{{Question:-}}}}$

The roots of a quadratic equation are 5 and -2. Find the equation.

$\huge {\bf{\orange{{Answer:-}}}}$

$\textsf{The roots of the quadratic equation are 5 and - 2.}$

$\boxed{\therefore{\sf{Factors \: of \: the \: quadratic \: equation \: will \: be \: (x \: - \: 5) \: and \: (x \: + \: 2).}}}$

$\textsf{By multiplying the factors we get the Quadratic equation,}$

$\sf{(x \: - \: 5) \: (x \: + \: 2)}$

$\sf{x \: (x \: + \: 2) \: - \: 5(x \: + \: 2)}$

$\sf{x^{2} \: + \: 2x \: - \: 5x \: - \: 10}$

$\boxed{\sf{x^{2} \: - \: 3x \: - \: 10}}$

$\huge{\bf{\red{{Conclusion:-}}}}$

$\boxed{\therefore{\sf{x^{2} \: - \: 3x \: - \: 10 \: is \: the \: quadratic \: equation \: whose \: roots \: are \: 5 \: and \: - \: 2.}}}$

$\huge{\bf{\pink{{Verification:-}}}}$

$\sf{x^{2} \: - \: 3x \: - \: 10 \: = \: 0}$

$\textsf{Substituting x = 5,}$

$\sf{5^{2} \: - 3 \: * \: 5 \: - \: 10 \: = \: 0}$

$\sf{25 \: - \: 15 \: - \: 10 \: = \: 0}$

$\sf{10 \: - \: 10 \: = \: 0}$

$\sf{0 \: = \: 0}$

$\textsf{Substituting x = - 2,}$

$\sf{- \: 2^{2} \: - 3 \: * \: - \: 2 \: - \: 10 \: = \: 0}$

$\sf{4 \: + \: 6 \: - \: 10 \: = \: 0}$

$\sf{10 \: - \: 10 \: = \: 0}$

$\sf{0 \: = \: 0}$

$\textsf{LHS = RHS in both the cases.}$

$\huge{\bf{\purple{{Note:-}}}}$

• If roots of a quadratic equation is given, then the factors of that quadratic equation can be found out by equating the roots to x.

• $\textsf{In the above problem,}$

• $\sf{5 \:= \:x}$

• x - 5 = 0 is the factor of the Quadratic equation.

• $\sf{- \:2 \:= \:x}$

• x + 2 = 0 is the factor of the Quadratic equation.

• Therefore, (x - 5) and (x + 2) are the factors of the Quadratic equation.

• Ignore ( = 0) as we require here only the factors.

Answer 2

Required quadratic equation