# Honors Algebra 2: Practice: Quadratic Systems

• Class: Honors Algebra 2
• Author: Peter Atlas
• Text: Algebra and Trigonometry: Structure and Method, Brown

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1. True or false: a system consisting of a linear and quadratic equation could have at most two real solutions.
2. Solution

True

3. True or false: a system consisting of two quadratic equations could never have exactly five real solutions.
4. Solution

True

5. True or false: the intersection of a circle and ellipse both with centers at the origin could contain eactly one point.
6. Solution

False

7. Solve the system over the real numbers: $$y = x^2 - 3$$ and $$y = 2x.$$
8. Solution

$$(3, 6)$$ and $$(-1, -2)$$

9. Solve the system over the real numbers: $$4y^2 - 25x^2 = 100$$ and $$y = 2x.$$
10. Solution

No real solutions

11. Find the dimensions of a rectangle whose area is 64 square cm and whose perimeter is 40 cm.
12. Solution

16 cm by 4 cm

13. Solve the system over the real numbers: $$x^2 + y^2 = 9$$ and $$y = x^2 - 3.$$
14. Solution

$$(0, -3),$$ $$(-\sqrt{5}, 2),$$ and $$(\sqrt{5}, 2).$$

15. Solve the system over the complex numbers: $$\displaystyle \frac{x^2}{9} + \frac{y^2}{4} = 1$$ and $$x^2 + y^2 =4.$$
16. Solution

$$(0, 2)$$ and $$(0, -2)$$

17. Solve the system over the complex numbers: $$\displaystyle \frac{x^2}{16} + \frac{y^2}{9} = 1$$ and $$\displaystyle \frac{x^2}{16} - \frac{y^2}{9} = 1.$$
18. Solution

$$(4, 0)$$ and $$(-4, 0)$$

19. Suppose you bought $1200 of stock. When the price per share increased by$1, you sold all but 10 shares for $1380. How much did you originally pay for each share of stock? 20. Solution$5

21. Solve over the complex numbers: $$9x^2 + y^2 = 27,$$ and $$y = 3x^2 - 3$$
22. Solution

$$( \pm i, -6)$$ and $$( \pm \sqrt{2}, 3)$$

23. If the numerator of a reduced, simplified fraction is increased by 3 and the denominator is decreased by 3, the resulting fraction is the reciprocal of the original fraction. The numerator of the original fraction is 1 more than one half its denominator. What was the original fraction?
24. Solution

$$\displaystyle \frac{5}{8}$$

25. The product of a two-digit number and the number obtained by reversing its digits is 2268. If the difference of the numbers is 27, find the numbers.
26. Solution

63