Mister Exam
Factor -8*y^2+12*y*x+7*x^2 squared
The solution
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2 2 - 8*y + 12*y*x + 7*x
$$7 x^{2} + \left(x 12 y - 8 y^{2}\right)$$
-8*y^2 + (12*y)*x + 7*x^2
General simplification
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2 2 - 8*y + 7*x + 12*x*y
$$7 x^{2} + 12 x y - 8 y^{2}$$
-8*y^2 + 7*x^2 + 12*x*y
The perfect square
Let's highlight the perfect square of the square three-member
$$7 x^{2} + \left(x 12 y - 8 y^{2}\right)$$
Let us write down the identical expression
$$7 x^{2} + \left(x 12 y - 8 y^{2}\right) = - \frac{92 y^{2}}{7} + \left(7 x^{2} + 12 x y + \frac{36 y^{2}}{7}\right)$$
or
$$7 x^{2} + \left(x 12 y - 8 y^{2}\right) = - \frac{92 y^{2}}{7} + \left(\sqrt{7} x + \frac{6 \sqrt{7} y}{7}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{92}{7}} y + \left(\sqrt{7} x + \frac{6 \sqrt{7}}{7} y\right)\right) \left(\sqrt{\frac{92}{7}} y + \left(\sqrt{7} x + \frac{6 \sqrt{7}}{7} y\right)\right)$$
$$\left(- \frac{2 \sqrt{161}}{7} y + \left(\sqrt{7} x + \frac{6 \sqrt{7}}{7} y\right)\right) \left(\frac{2 \sqrt{161}}{7} y + \left(\sqrt{7} x + \frac{6 \sqrt{7}}{7} y\right)\right)$$
$$\left(\sqrt{7} x + y \left(- \frac{2 \sqrt{161}}{7} + \frac{6 \sqrt{7}}{7}\right)\right) \left(\sqrt{7} x + y \left(\frac{6 \sqrt{7}}{7} + \frac{2 \sqrt{161}}{7}\right)\right)$$
$$\left(\sqrt{7} x + y \left(- \frac{2 \sqrt{161}}{7} + \frac{6 \sqrt{7}}{7}\right)\right) \left(\sqrt{7} x + y \left(\frac{6 \sqrt{7}}{7} + \frac{2 \sqrt{161}}{7}\right)\right)$$
$$7 x^{2} + \left(x 12 y - 8 y^{2}\right)$$
Let us write down the identical expression
$$7 x^{2} + \left(x 12 y - 8 y^{2}\right) = - \frac{92 y^{2}}{7} + \left(7 x^{2} + 12 x y + \frac{36 y^{2}}{7}\right)$$
or
$$7 x^{2} + \left(x 12 y - 8 y^{2}\right) = - \frac{92 y^{2}}{7} + \left(\sqrt{7} x + \frac{6 \sqrt{7} y}{7}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{92}{7}} y + \left(\sqrt{7} x + \frac{6 \sqrt{7}}{7} y\right)\right) \left(\sqrt{\frac{92}{7}} y + \left(\sqrt{7} x + \frac{6 \sqrt{7}}{7} y\right)\right)$$
$$\left(- \frac{2 \sqrt{161}}{7} y + \left(\sqrt{7} x + \frac{6 \sqrt{7}}{7} y\right)\right) \left(\frac{2 \sqrt{161}}{7} y + \left(\sqrt{7} x + \frac{6 \sqrt{7}}{7} y\right)\right)$$
$$\left(\sqrt{7} x + y \left(- \frac{2 \sqrt{161}}{7} + \frac{6 \sqrt{7}}{7}\right)\right) \left(\sqrt{7} x + y \left(\frac{6 \sqrt{7}}{7} + \frac{2 \sqrt{161}}{7}\right)\right)$$
$$\left(\sqrt{7} x + y \left(- \frac{2 \sqrt{161}}{7} + \frac{6 \sqrt{7}}{7}\right)\right) \left(\sqrt{7} x + y \left(\frac{6 \sqrt{7}}{7} + \frac{2 \sqrt{161}}{7}\right)\right)$$
Factorization
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/ / ____\\ / / ____\\ | 2*y*\-3 + \/ 23 /| | 2*y*\3 + \/ 23 /| |x - -----------------|*|x + ----------------| \ 7 / \ 7 /
$$\left(x - \frac{2 y \left(-3 + \sqrt{23}\right)}{7}\right) \left(x + \frac{2 y \left(3 + \sqrt{23}\right)}{7}\right)$$
(x - 2*y*(-3 + sqrt(23))/7)*(x + 2*y*(3 + sqrt(23))/7)
Combining rational expressions
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2 7*x + 4*y*(-2*y + 3*x)
$$7 x^{2} + 4 y \left(3 x - 2 y\right)$$
7*x^2 + 4*y*(-2*y + 3*x)
Assemble expression
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2 2 - 8*y + 7*x + 12*x*y
$$7 x^{2} + 12 x y - 8 y^{2}$$
-8*y^2 + 7*x^2 + 12*x*y
Rational denominator
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2 2 - 8*y + 7*x + 12*x*y
$$7 x^{2} + 12 x y - 8 y^{2}$$
-8*y^2 + 7*x^2 + 12*x*y
Common denominator
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2 2 - 8*y + 7*x + 12*x*y
$$7 x^{2} + 12 x y - 8 y^{2}$$
-8*y^2 + 7*x^2 + 12*x*y
Trigonometric part
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2 2 - 8*y + 7*x + 12*x*y
$$7 x^{2} + 12 x y - 8 y^{2}$$
-8*y^2 + 7*x^2 + 12*x*y