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3x^{2}+1
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3x^{2}+1
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1^{2}-\left(2x\right)^{2}+\left(1-3x\right)\left(3x+1\right)-\left(4x+1\right)\left(1-4x\right)
Consider \left(1-2x\right)\left(1+2x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
1-\left(2x\right)^{2}+\left(1-3x\right)\left(3x+1\right)-\left(4x+1\right)\left(1-4x\right)
Calculate 1 to the power of 2 and get 1.
1-2^{2}x^{2}+\left(1-3x\right)\left(3x+1\right)-\left(4x+1\right)\left(1-4x\right)
Expand \left(2x\right)^{2}.
1-4x^{2}+\left(1-3x\right)\left(3x+1\right)-\left(4x+1\right)\left(1-4x\right)
Calculate 2 to the power of 2 and get 4.
1-4x^{2}+1^{2}-\left(3x\right)^{2}-\left(4x+1\right)\left(1-4x\right)
Consider \left(1-3x\right)\left(3x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
1-4x^{2}+1-\left(3x\right)^{2}-\left(4x+1\right)\left(1-4x\right)
Calculate 1 to the power of 2 and get 1.
1-4x^{2}+1-3^{2}x^{2}-\left(4x+1\right)\left(1-4x\right)
Expand \left(3x\right)^{2}.
1-4x^{2}+1-9x^{2}-\left(4x+1\right)\left(1-4x\right)
Calculate 3 to the power of 2 and get 9.
2-4x^{2}-9x^{2}-\left(4x+1\right)\left(1-4x\right)
Add 1 and 1 to get 2.
2-13x^{2}-\left(4x+1\right)\left(1-4x\right)
Combine -4x^{2} and -9x^{2} to get -13x^{2}.
2-13x^{2}-\left(1^{2}-\left(4x\right)^{2}\right)
Consider \left(4x+1\right)\left(1-4x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
2-13x^{2}-\left(1-\left(4x\right)^{2}\right)
Calculate 1 to the power of 2 and get 1.
2-13x^{2}-\left(1-4^{2}x^{2}\right)
Expand \left(4x\right)^{2}.
2-13x^{2}-\left(1-16x^{2}\right)
Calculate 4 to the power of 2 and get 16.
2-13x^{2}-1-\left(-16x^{2}\right)
To find the opposite of 1-16x^{2}, find the opposite of each term.
2-13x^{2}-1+16x^{2}
The opposite of -16x^{2} is 16x^{2}.
1-13x^{2}+16x^{2}
Subtract 1 from 2 to get 1.
1+3x^{2}
Combine -13x^{2} and 16x^{2} to get 3x^{2}.
1^{2}-\left(2x\right)^{2}+\left(1-3x\right)\left(3x+1\right)-\left(4x+1\right)\left(1-4x\right)
Consider \left(1-2x\right)\left(1+2x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
1-\left(2x\right)^{2}+\left(1-3x\right)\left(3x+1\right)-\left(4x+1\right)\left(1-4x\right)
Calculate 1 to the power of 2 and get 1.
1-2^{2}x^{2}+\left(1-3x\right)\left(3x+1\right)-\left(4x+1\right)\left(1-4x\right)
Expand \left(2x\right)^{2}.
1-4x^{2}+\left(1-3x\right)\left(3x+1\right)-\left(4x+1\right)\left(1-4x\right)
Calculate 2 to the power of 2 and get 4.
1-4x^{2}+1^{2}-\left(3x\right)^{2}-\left(4x+1\right)\left(1-4x\right)
Consider \left(1-3x\right)\left(3x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
1-4x^{2}+1-\left(3x\right)^{2}-\left(4x+1\right)\left(1-4x\right)
Calculate 1 to the power of 2 and get 1.
1-4x^{2}+1-3^{2}x^{2}-\left(4x+1\right)\left(1-4x\right)
Expand \left(3x\right)^{2}.
1-4x^{2}+1-9x^{2}-\left(4x+1\right)\left(1-4x\right)
Calculate 3 to the power of 2 and get 9.
2-4x^{2}-9x^{2}-\left(4x+1\right)\left(1-4x\right)
Add 1 and 1 to get 2.
2-13x^{2}-\left(4x+1\right)\left(1-4x\right)
Combine -4x^{2} and -9x^{2} to get -13x^{2}.
2-13x^{2}-\left(1^{2}-\left(4x\right)^{2}\right)
Consider \left(4x+1\right)\left(1-4x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
2-13x^{2}-\left(1-\left(4x\right)^{2}\right)
Calculate 1 to the power of 2 and get 1.
2-13x^{2}-\left(1-4^{2}x^{2}\right)
Expand \left(4x\right)^{2}.
2-13x^{2}-\left(1-16x^{2}\right)
Calculate 4 to the power of 2 and get 16.
2-13x^{2}-1-\left(-16x^{2}\right)
To find the opposite of 1-16x^{2}, find the opposite of each term.
2-13x^{2}-1+16x^{2}
The opposite of -16x^{2} is 16x^{2}.
1-13x^{2}+16x^{2}
Subtract 1 from 2 to get 1.
1+3x^{2}
Combine -13x^{2} and 16x^{2} to get 3x^{2}.
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