Honors Algebra 2: Extra Practice on Discriminants and Sum and Product of the Roots.

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

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1. Determine the nature of the roots of $$\displaystyle 3x^2 + 2x - 6 = 0$$
2. Solution

2 real irrational roots

3. Determine the nature of the roots of $$\displaystyle x^2 - 2\sqrt{5}x + 5 = 0$$
4. Solution

one real rational (double) root

5. Find the values of $$b$$ and $$c$$ if the equation $$\displaystyle 4x^2 + bx + c = 0$$ has as its roots $$\displaystyle \frac{1 + \sqrt{2}}{2},$$ and $$\displaystyle \frac{1 - \sqrt{2}}{2}.$$
6. Solution

$$\displaystyle b = -4; c = -1$$

7. Find a quadratic equation whose roots are $$\displaystyle 1 + i\sqrt{3}$$ and $$\displaystyle 1 - i\sqrt{3}$$
8. Solution

$$\displaystyle x^2 - 2x + 4 = 0$$ or any scalar multiple thereof. (That is, I'd accept anything in the form $$\displaystyle a(x^2 - 2x + 4) = 0$$ for any value of $$a.$$

9. Determine the value(s) of $$k$$ for which the given equation will have exactly one real root: $$\displaystyle 5x^2 - kx + 8 = 0$$
10. Solution

$$\displaystyle k = \pm 4\sqrt{10}$$

11. Determine the value(s) of $$k$$ for which the given equation will have imaginary roots. $$\displaystyle x^2 + 4(k + 1)x + 4k^2 = 0$$
12. Solution

$$\displaystyle k < -\frac{1}{2}$$

13. Determine the value(s) of $$k$$ so that one root of $$16x^2 - 9x + k = 0$$ is three times the other.
14. Solution

$$\displaystyle \frac{243}{256}$$