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The unique pair of values a, b satisfying the first two equations is (a, b) = (1, 1); since these values also satisfy the third equation, there do in fact exist a, b such that a times the original first equation plus b times the original second equation equals the original third equation; we conclude that the third equation is linearly ...
To complete the square, form a squared binomial on the left-hand side of a quadratic equation, from which the solution can be found by taking the square root of both sides. The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation a x 2 + b x + c = 0 {\displaystyle ...
This equation has the same two solutions as the original one: = and = We can also modify the solution set by squaring both sides, because this will make any negative values in the ranges of the equation positive, causing extraneous solutions.
Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes the square", converting the left side into a perfect square. Write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of ...
In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with some linear operator L on the LHS and 0 on the RHS. In contrast, an equation with a non-zero RHS is called inhomogeneous or non-homogeneous, as exemplified by ...
For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity. If some function is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called extraneous solutions .
If >, then the equation = + + describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the ordinate of the minimum point of the corresponding parabola y p = a x 2 + b x + c . {\displaystyle y_{p}=ax^{2}+bx+c.}
Squaring both sides of the equation gives ... Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the ...