Solving quadratic equations, algebra homework help

I have the answers and explanation below. Please elaborate and explain how to solve the questions. I don’t understand exactly the process of factoring them and then simplifying. Please help.

Factor.
3h? + 7h + 4
You answered:
To factor a quadratic of the form ax? + bx + C, write it as
ax? + r1x + r2x + C
where a . c = ri.ro and b = rı + r2. Then factor by grouping.
Look at the given quadratic:
3h? + 7h + 4
The product acis 12, so you need to find a pair of factors with a product of 12. The b term
is 7, so you need to find a pair of factors with a sum of 7. Since the product is positive (12)
and the sum is positive (7), you need both factors to be positive.
Make a list of the possible factor pairs with a product of 12, and then find the one with a sum
of 7.
Factor pairs of a . c = 12 Sum of factor pairs
1.12 = 12
1 + 12 = 13
2.6 = 12
2 + 6 = 8
3.4 = 12
3 + 4 = 7
The factors 3 and 4 have a sum of 7. So, replace the quadratic’s 7h term with 3h and 4h,
and then factor by grouping.
3h2 + 7h + 4
3h2 + 3h + 4h + 4
3h(h + 1) + 4h + 1)
Factor by grouping; the expressions in parentheses should match
(3h + 4)( + 1)
f2 + 8f + 16 = 0
Write your answers as integers or as proper or improper fractions in simplest form. If there
are multiple answers, separate them with commas.
f=
You answered:
f =
To factor a quadratic of the form x? + bx + C, write it as
(x + r.)(x + 2)
where c = rir and b = r1 + r2.
Step 1: Factor.
The c term is 16, so you need to find a pair of factors with a product of 16. The b term is 8,
so you need to find a pair of factors with a sum of 8. Since the product is positive (16) and
the sum is positive (8), you need both factors to be positive.
Make a list of the possible factor pairs with a product of 16, and then find the one with a sum
of 8.
Factor pairs of c = 16
Sum of factor pairs
1.16 = 16
1 + 16 = 17
2.8 = 16
2 + 8 = 10
4.4 = 16
4 + 4 = 8
The factors 4 and 4 have a sum of 8. Use those numbers to factor the quadratic inside the
parentheses, f? + 8f + 16.
f2 + 8f + 16 = 0
(f + 4) (f + 4) = 0
Step 2: Use the zero product property to solve.
According to the zero product property, if (f + 4)(f + 4) = 0, then f + 4 must be o orf + 4
must be 0. Write the two equations and solve for f.
f + 4 = 0 or
f + 4 = 0
f = -4
f = -4
22 + 18z – 13 = 0
Write your answers as integers, proper or improper fractions in simplest form, or decimals
rounded to the nearest hundredth.
Z =
or z =
You answered:
z =
or z =
With quadratic equations (ax2 + bx + c = 0), you can solve by completing the square.
Step 1: Make sure that the left side of the equation looks like x2 + bx.
Step 2: Add
()
to both sides.
Step 3: Factor the left side as x +
(x+)?
Step 4: Take the square root and solve.
Step 1: Make sure that the left side of the equation looks like x2 + bx.
To make the left side of the equation look like x2 + bx, add 13 to both sides.
22 + 18z – 13 = 0
22 + 18z = 13
b
Step 2: Add
to both sides.
Since b = 18,
*) * = ()
= 92 = 81. Add 81 to both sides.
22 + 18z + 81 = 94
Step 3: Factor the left side as x +
)?
In general, an expression of the form x2 + bx +
()
can be factored as
(x+)?
+
2
In general, an expression of the form x2 + bx +
can be factored as x +
NO
The expression z2 + 18z + 81 is of this form, with b = 18. So, it can be factored as
(z + 9)2
Rewrite the equation with the left side factored.
(z + 9)2 = 94
Step 4: Take the square root and solve.
z +92 +9.70
Take the square root
Z-9 + 9.70
Subtract 9 from both sides
Z-9 + 9.70 or Z-9-9.70
Split = into + or –
Z 0.70 or z-18.70
Simplify
Solve using the quadratic formula.
4y2 – 6y + 1 = 0
Write your answers as integers, proper or improper fractions in simplest form, or decimals
rounded to the nearest hundredth.
y =
or y =
You answered:
y =
or y =
The quadratic formula is
х
-b + b2 – 4ac
2a
where a 0. It can be used to sol
a quadratic equation (ax2 + bx + c = 0).
Use the quadratic formula to solve 4y2 – 6y + 1 = 0.
y =
-b + b2 – 4ac
2a
y =
-(-6) + (-6)2 – 4(4)(1)
2(4)
Plug in a = 4, b = -6, and c = 1
6 +
y =
36 – 16
8
Multiply
6 + 20
y =
8
Subtract
6 + 20
y =
8
6
or y =
20
8
Split into + or –
y 1.31 or y = 0.19
Simplify and round to the nearest hundredth