Question

Can someone help me with my practice test?

Answer 1

9. (x +3)² +4x = x² +10x +9

10. (5 +3m)/15

11. x = (-1±i√14)/3

Explanation:

You want help with practice test questions involving quadratics and rational numbers.

9. Equivalence

For this, it is perhaps easiest to show equivalence by multiplying out the second form of the quadratic:

(x +3)² +4x

= (x +3)(x +3) +4x

= x(x +3) +3(x +3) +4x

= x² +3x +3x +9 +4x

= x² +(3 +3 +4)x +9

= x² +10x +9 . . . . . . . . equivalent form

10. Adding rationals

As you know, fractions can be added by adding their numerators when they have a common denominator. Here, the denominators of 3 and 5 have no common factors, so the least common denominator is their product, 3·5 = 15. Writing equivalent fractions with denominators of 15, we have ...

`(1)/(3)+(m)/(5)=(5)/(5)\cdot(1)/(3)+(3)/(3)\cdot(m)/(5)\\\\=(5)/(15)+(3m)/(15)=\boxed{(5+3m)/(15)}`

It can be worthwhile to remember that you can always add fractions with the formula ...

`(a)/(b)+(c)/(d)=(ad+bc)/(bd)`

11. Quadratic formula

The solution to 3x² +2x +5 = 0 is found using the quadratic formula with ...

• a = 3
• b = 2
• c = 5

`x=(-b\pm√(b^2-4ac))/(2a)=(-2\pm√(2^2-4\cdot3\cdot5))/(2\cdot3)\\\\=(-2\pm√(-56))/(6)=(-2\pm2i√(14))/(6)\\\\\boxed{x=(-1\pm i√(14))/(3)}`

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Additional comment

It can be useful to remember ...

(a ±b)² = a² ±2ab +b²

where the sign of the middle term matches the sign in the binomial. This is also used for "completing the square" where the b² term is the square of the middle term after it is divided by 2a.