Question

1. look back at each “row” of factored expressions in problems 14 to 25 above. explain how it is possible that the coefficient (b) of the middle term can be different numbers in each problem when the “outside” coefficients (a) and (c) are the same. (recall the standard form of a quadratic is ax² + bx + c.)

go
topic: taking the square root of perfect squares.
only some of the expressions inside the radical sign are perfect squares. identify which ones are perfect squares and take the square root. leave the ones that are not perfect squares under the radical sign. do not attempt to simplify them. (hint: check your answers by squaring them. you should be able to get what you started with, if you are right.)

1. $sqrt{(17xyz)^2}$
2. $sqrt{(3x - 7)^2}$
3. $sqrt{121a^{2}b^{6}}$
4. $sqrt{x^{2}+8x + 16}$
5. $sqrt{x^{2}+14x + 49}$
6. $sqrt{x^{2}+14x - 49}$
7. $sqrt{x^{2}+10x + 100}$
8. $sqrt{x^{2}+20x + 100}$
9. $sqrt{x^{2}-20x + 100}$

Explanation:

Step1: Recall square - root property

For $\sqrt{a^{2}}=\vert a\vert$.

Step2: Solve 27

$\sqrt{(17xyz)^{2}}=\vert17xyz\vert = 17xyz$ (assuming $x,y,z$ are real - valued and non - negative for simplicity).

Step3: Solve 28

$\sqrt{(3x - 7)^{2}}=\vert3x - 7\vert$.

Step4: Solve 29

$\sqrt{121a^{2}b^{6}}=\sqrt{11^{2}(a)^{2}(b^{3})^{2}} = 11\vert a\vert b^{3}$.

Step5: Solve 30

$x^{2}+8x + 16=(x + 4)^{2}$, so $\sqrt{x^{2}+8x + 16}=\vert x + 4\vert$.

Step6: Solve 31

$x^{2}+14x + 49=(x + 7)^{2}$, so $\sqrt{x^{2}+14x + 49}=\vert x+7\vert$.

Step7: Solve 32

$x^{2}+14x - 49$ is not a perfect square, so leave it as $\sqrt{x^{2}+14x - 49}$.

Step8: Solve 33

$x^{2}+10x + 100$ is not a perfect square, so leave it as $\sqrt{x^{2}+10x + 100}$.

Step9: Solve 34

$x^{2}+20x + 100=(x + 10)^{2}$, so $\sqrt{x^{2}+20x + 100}=\vert x + 10\vert$.

Step10: Solve 35

$x^{2}-20x + 100=(x - 10)^{2}$, so $\sqrt{x^{2}-20x + 100}=\vert x - 10\vert$.

Answer:

1. $17xyz$
2. $\vert3x - 7\vert$
3. $11\vert a\vert b^{3}$
4. $\vert x + 4\vert$
5. $\vert x+7\vert$
6. $\sqrt{x^{2}+14x - 49}$
7. $\sqrt{x^{2}+10x + 100}$
8. $\vert x + 10\vert$
9. $\vert x - 10\vert$