Mister Exam
Factor -9*y^2+5*y*q+8*q^2 squared
The solution
You have entered
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2 2 - 9*y + 5*y*q + 8*q
$$8 q^{2} + \left(q 5 y - 9 y^{2}\right)$$
-9*y^2 + (5*y)*q + 8*q^2
General simplification
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2 2 - 9*y + 8*q + 5*q*y
$$8 q^{2} + 5 q y - 9 y^{2}$$
-9*y^2 + 8*q^2 + 5*q*y
The perfect square
Let's highlight the perfect square of the square three-member
$$8 q^{2} + \left(q 5 y - 9 y^{2}\right)$$
Let us write down the identical expression
$$8 q^{2} + \left(q 5 y - 9 y^{2}\right) = - \frac{313 y^{2}}{32} + \left(8 q^{2} + 5 q y + \frac{25 y^{2}}{32}\right)$$
or
$$8 q^{2} + \left(q 5 y - 9 y^{2}\right) = - \frac{313 y^{2}}{32} + \left(2 \sqrt{2} q + \frac{5 \sqrt{2} y}{8}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{313}{32}} y + \left(2 \sqrt{2} q + \frac{5 \sqrt{2}}{8} y\right)\right) \left(\sqrt{\frac{313}{32}} y + \left(2 \sqrt{2} q + \frac{5 \sqrt{2}}{8} y\right)\right)$$
$$\left(- \frac{\sqrt{626}}{8} y + \left(2 \sqrt{2} q + \frac{5 \sqrt{2}}{8} y\right)\right) \left(\frac{\sqrt{626}}{8} y + \left(2 \sqrt{2} q + \frac{5 \sqrt{2}}{8} y\right)\right)$$
$$\left(2 \sqrt{2} q + y \left(\frac{5 \sqrt{2}}{8} + \frac{\sqrt{626}}{8}\right)\right) \left(2 \sqrt{2} q + y \left(- \frac{\sqrt{626}}{8} + \frac{5 \sqrt{2}}{8}\right)\right)$$
$$\left(2 \sqrt{2} q + y \left(\frac{5 \sqrt{2}}{8} + \frac{\sqrt{626}}{8}\right)\right) \left(2 \sqrt{2} q + y \left(- \frac{\sqrt{626}}{8} + \frac{5 \sqrt{2}}{8}\right)\right)$$
$$8 q^{2} + \left(q 5 y - 9 y^{2}\right)$$
Let us write down the identical expression
$$8 q^{2} + \left(q 5 y - 9 y^{2}\right) = - \frac{313 y^{2}}{32} + \left(8 q^{2} + 5 q y + \frac{25 y^{2}}{32}\right)$$
or
$$8 q^{2} + \left(q 5 y - 9 y^{2}\right) = - \frac{313 y^{2}}{32} + \left(2 \sqrt{2} q + \frac{5 \sqrt{2} y}{8}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{313}{32}} y + \left(2 \sqrt{2} q + \frac{5 \sqrt{2}}{8} y\right)\right) \left(\sqrt{\frac{313}{32}} y + \left(2 \sqrt{2} q + \frac{5 \sqrt{2}}{8} y\right)\right)$$
$$\left(- \frac{\sqrt{626}}{8} y + \left(2 \sqrt{2} q + \frac{5 \sqrt{2}}{8} y\right)\right) \left(\frac{\sqrt{626}}{8} y + \left(2 \sqrt{2} q + \frac{5 \sqrt{2}}{8} y\right)\right)$$
$$\left(2 \sqrt{2} q + y \left(\frac{5 \sqrt{2}}{8} + \frac{\sqrt{626}}{8}\right)\right) \left(2 \sqrt{2} q + y \left(- \frac{\sqrt{626}}{8} + \frac{5 \sqrt{2}}{8}\right)\right)$$
$$\left(2 \sqrt{2} q + y \left(\frac{5 \sqrt{2}}{8} + \frac{\sqrt{626}}{8}\right)\right) \left(2 \sqrt{2} q + y \left(- \frac{\sqrt{626}}{8} + \frac{5 \sqrt{2}}{8}\right)\right)$$
Factorization
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/ / _____\\ / / _____\\ | y*\-5 + \/ 313 /| | y*\5 + \/ 313 /| |q - ----------------|*|q + ---------------| \ 16 / \ 16 /
$$\left(q - \frac{y \left(-5 + \sqrt{313}\right)}{16}\right) \left(q + \frac{y \left(5 + \sqrt{313}\right)}{16}\right)$$
(q - y*(-5 + sqrt(313))/16)*(q + y*(5 + sqrt(313))/16)
Trigonometric part
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2 2 - 9*y + 8*q + 5*q*y
$$8 q^{2} + 5 q y - 9 y^{2}$$
-9*y^2 + 8*q^2 + 5*q*y
Assemble expression
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2 2 - 9*y + 8*q + 5*q*y
$$8 q^{2} + 5 q y - 9 y^{2}$$
-9*y^2 + 8*q^2 + 5*q*y
Combining rational expressions
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2 8*q + y*(-9*y + 5*q)
$$8 q^{2} + y \left(5 q - 9 y\right)$$
8*q^2 + y*(-9*y + 5*q)
Rational denominator
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2 2 - 9*y + 8*q + 5*q*y
$$8 q^{2} + 5 q y - 9 y^{2}$$
-9*y^2 + 8*q^2 + 5*q*y
Common denominator
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2 2 - 9*y + 8*q + 5*q*y
$$8 q^{2} + 5 q y - 9 y^{2}$$
-9*y^2 + 8*q^2 + 5*q*y