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→Examples: completing the cases |
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:<math>Q(x)=c_1{x_1}^2+c_2{x_2}^2 </math>
where {{math|1=''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>)}} and {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are constants. If {{math|1=''c''<sub>1</sub> > 0}} and {{math|1=''c''<sub>2</sub> > 0}}, the quadratic form {{math|''Q''}} is positive definite, so ''Q'' evaluates to a positive number whenever <math>(x_1,x_2)\neq (0,0).</math> If one of the constants is positive and the other is 0, then {{math|''Q''}} is positive semidefinite and always evaluates to either 0 or a positive number. If {{math|1=''c''<sub>1</sub> > 0}} and {{math|1=''c''<sub>2</sub> < 0}}, or vice versa, then {{math|''Q''}} is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If {{math|1=''c''<sub>1</sub> < 0}} and {{math|1=''c''<sub>2</sub> < 0}}, the quadratic form is negative definite and always evaluates to a negative number whenever <math>(x_1,x_2)\neq (0,0).</math> And if one of the constants is negative and the other is 0, then {{math|''Q''}} is negative semidefinite and always evaluates to either 0 or a negative number.
In general a quadratic form in two variables will also involve a cross-product term in ''x''<sub>1</sub>''x''<sub>2</sub>:
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