Your friend concluded that because two discriminants are equal, the solutions to the two equations are the same. Explain your friend's error. Give an example of two quadratic equations that disprove this conclusion.

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Ответ:

enicole10

04.04.2021

Mathematics

$\boxed{ \ (x, y) \rightarrow (-y, x) \ }$

Further explanation

We already know there are four types of transformation geometry: Translation, Reflection, Rotation, and Dilation.

Rotation represents a type of transformation in which a figure is rotated about fixed points called centers of rotation. The result of rotation is the same shape and size, but the figures may be turned in different directions. Rotations may be clockwise or counterclockwise. A positive angle of rotation moves the figure counterclockwise, and a negative angle turns the figure clockwise. Without being mentioned, it means a positive angle.

However, we must moreover know about the use of multiplier matrices in transformation geometry. The matrix used in rotation 90° in the counterclockwise direction, in other words, a positive angle 90°, is

$\left[\begin{array}{ccc}0&-1\\1&0\\\end{array}\right]$

State (x, y) as the initial coordinate and (x', y') as the final coordinate.

The results of rotation are obtained from the multiplication of the matrix with the initial coordinates.

$\left[\begin{array}{ccc}x'\\y'\\\end{array}\right] = \left[\begin{array}{ccc}0&-1\\1&0\\\end{array}\right]\left[\begin{array}{ccc}x\\y\\\end{array}\right]$

$\left[\begin{array}{ccc}x'\\y'\\\end{array}\right] = \left[\begin{array}{ccc}(0)(x) + (-1)(y)\\(1)(x) + (0)(y)\\\end{array}\right]$

$\left[\begin{array}{ccc}x'\\y'\\\end{array}\right] = \left[\begin{array}{ccc}-y\\x\\\end{array}\right]$

It can be concluded that if a point is rotated 90° about the origin, the rule that describes the transformation is $\boxed{ \ (x, y) \rightarrow (-y, x) \ }$

Example:

Triangle ABC with coordinates of points A (3, 2), B (6, 2), and C (3, 6) is rotated 90 about the origin. The points of rotation results are called A', B', and C'.

Let's find out the rotation results called A', B', and C' implementing the rule above.

$\boxed{ \ A(3, 2) \rightarrow A'(-2, 3) \ }$

$\boxed{ \ B(6, 2) \rightarrow B'(-2, 6) \ }$

$\boxed{ \ C(3, 6) \rightarrow C'(-6, 3) \ }$

Learn moreThe similar problem A dilation is NOT rigid motion transformation The type of transformation applied in figure

Keywords: triangle, rotated, rotation, 90 degrees, about the origin, transformation geometry, translation, reflection, dilation, multiplier matrix, initial coordinate, rule, describes

Atriangle is rotated 90° about the origin. which rule describes the transformation? (x, y) → (–x, –

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Your friend concluded that because two discriminants are equal, the solutions to the two equations are the same. Explain your friend's error. Give an example of two quadratic equations that disprove this conclusion.​