Mister Exam
Factor -9*a^2-6*a*b+7*b^2 squared
The solution
You have entered
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2 2 - 9*a - 6*a*b + 7*b
$$7 b^{2} + \left(- 9 a^{2} - 6 a b\right)$$
-9*a^2 - 6*a*b + 7*b^2
General simplification
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2 2 - 9*a + 7*b - 6*a*b
$$- 9 a^{2} - 6 a b + 7 b^{2}$$
-9*a^2 + 7*b^2 - 6*a*b
Factorization
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/ / ___\\ / / ___\\ | b*\-1 + 2*\/ 2 /| | b*\1 + 2*\/ 2 /| |a - ----------------|*|a + ---------------| \ 3 / \ 3 /
$$\left(a - \frac{b \left(-1 + 2 \sqrt{2}\right)}{3}\right) \left(a + \frac{b \left(1 + 2 \sqrt{2}\right)}{3}\right)$$
(a - b*(-1 + 2*sqrt(2))/3)*(a + b*(1 + 2*sqrt(2))/3)
The perfect square
Let's highlight the perfect square of the square three-member
$$7 b^{2} + \left(- 9 a^{2} - 6 a b\right)$$
Let us write down the identical expression
$$7 b^{2} + \left(- 9 a^{2} - 6 a b\right) = 8 b^{2} + \left(- 9 a^{2} - 6 a b - b^{2}\right)$$
or
$$7 b^{2} + \left(- 9 a^{2} - 6 a b\right) = 8 b^{2} - \left(3 a + b\right)^{2}$$
$$7 b^{2} + \left(- 9 a^{2} - 6 a b\right)$$
Let us write down the identical expression
$$7 b^{2} + \left(- 9 a^{2} - 6 a b\right) = 8 b^{2} + \left(- 9 a^{2} - 6 a b - b^{2}\right)$$
or
$$7 b^{2} + \left(- 9 a^{2} - 6 a b\right) = 8 b^{2} - \left(3 a + b\right)^{2}$$
Rational denominator
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2 2 - 9*a + 7*b - 6*a*b
$$- 9 a^{2} - 6 a b + 7 b^{2}$$
-9*a^2 + 7*b^2 - 6*a*b
Combining rational expressions
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2 7*b + 3*a*(-3*a - 2*b)
$$3 a \left(- 3 a - 2 b\right) + 7 b^{2}$$
7*b^2 + 3*a*(-3*a - 2*b)
Assemble expression
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2 2 - 9*a + 7*b - 6*a*b
$$- 9 a^{2} - 6 a b + 7 b^{2}$$
-9*a^2 + 7*b^2 - 6*a*b
Common denominator
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2 2 - 9*a + 7*b - 6*a*b
$$- 9 a^{2} - 6 a b + 7 b^{2}$$
-9*a^2 + 7*b^2 - 6*a*b
Trigonometric part
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2 2 - 9*a + 7*b - 6*a*b
$$- 9 a^{2} - 6 a b + 7 b^{2}$$
-9*a^2 + 7*b^2 - 6*a*b