5 exercises · 5 automatically checked · Untimed
5 questions · Shuffled order · Untimed practice
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If 4x − 3y = 11, what is 8x − 6y + 7?
Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.
29
The target is 2(4x − 3y) + 7. Replace the known group by 11 to obtain 2(11) + 7
What is the value of (10032 − 9972)/2000?
Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.
6
Apply the difference-of-squares identity: (10032 − 9972)/2000 = (1003 − 997)(1003 + 997)/2000 = 6(2000)/2000 = 6. The large intermediate squares never need to be calculated. See Example 3.04.
Real numbers a, b satisfy a + b = 12 and ab = 20. Find a2 + b2.
Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.
104
If the numbers are a and b, then a2 + b2 = (a + b)2 − 2ab = 122 − 2(20) = 104. The term 2ab is present in the square of a sum; leaving it out would give 144 instead. See Example 3.09.
The quadratic 2x2 − 10x + 3 = 0 has roots r, s. Find 1/r + 1/s.
Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.
10/3
For 2x2 − 10x + 3
For f(x) = x2 + 4x + 7, distinct real numbers p, q satisfy f (p) = f (q). Find p + q.
Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.
−4
Equal outputs give p2 + 4p + 7
Review Recognizing useful algebraic structure