Question

Question 1
Next Question
Rewrite the polynomial in the form a, x, squared, plus, b, x, plus, cax
2
+bx+c and then identify the values of a, commaa, b, commab, and cc.
minus, 9, plus, x, squared
−9+x
2Question 2
Next Question
Rewrite the polynomial in the form a, x, squared, plus, b, x, plus, cax
2
+bx+c and then identify the values of a, commaa, b, commab, and cc.
1, minus, one sixth, x, squared, minus, x
1−
6
1

x
2
−x
Your AnswerQuestion 3
Next Question
Rewrite the polynomial in the form a, x, squared, plus, b, x, plus, cax
2
+bx+c and then identify the values of a, commaa, b, commab, and cc.
3, x, plus, one quarter, x, squared
3x+
4
1

x
2
Question 4
Next Question
What is the discriminant of the quadratic equation 6, x, squared, minus, 2, x, minus, 1, equals, 06x
2
−2x−1=0?Question 5
Next Question
What is the discriminant of the quadratic equation minus, 2, x, squared, minus, 4, x, plus, 7, equals, 0−2x
2
−4x+7=0?Question 6
Next Question
What is the discriminant of the quadratic equation minus, 9, x, squared, minus, x, minus, 5, equals, 0−9x
2
−x−5=0?Question 7
Next Question
Use the quadratic formula to solve. Express your answer in simplest form.
8, z, squared, plus, 18, z, minus, 7, equals, minus, 2
8z
2
+18z−7=
−2Question 8
Next Question
Use the quadratic formula to solve. Express your answer in simplest form.
3, y, squared, minus, 8, y, plus, 4, equals, 5, y
3y
2
−8y+4=
5y
AnswerQuestion 9
Next Question
Use the quadratic formula to solve. Express your answer in simplest form.
16, c, squared, minus, 24, c, plus, 4, equals, minus, 1
16c
2
−24c+4=
−1
AnswerQuestion 10
Next Question
Solve the equation x, squared, plus, 16, x, plus, 56, equals, 0x
2
+16x+56=0 to the nearest tenth.Question 11
Next Question
Solve the equation 3, x, squared, minus, 8, x, minus, 14, equals, 03x
2
−8x−14=0 to the nearest tenth.Question 12
Next Question
Solve the equation 5, x, squared, minus, 11, x, minus, 7, equals, 05x
2
−11x−7=0 to the nearest tenth.Question 13
Next Question
If using the method of completing the square to solve the quadratic equation x, squared, plus, 3, x, plus, 37, equals, 0x
2
+3x+37=0, which number would have to be added to "complete the square"?Question 14
Next Question
If using the method of completing the square to solve the quadratic equation x, squared, plus, 16, x, minus, 5, equals, 0x
2
+16x−5=0, which number would have to be added to "complete the square"?
AnswerQuestion 15
Next Question
If using the method of completing the square to solve the quadratic equation x, squared, plus, 13, x, minus, 3, equals, 0x
2
+13x−3=0, which number would have to be added to "complete the square"?
AnswerQuestion 16
Next Question
If using the method of completing the square to solve the quadratic equation x, squared, minus, 19, x, minus, 24, equals, 0x
2
−19x−24=0, which number would have to be added to "complete the square"?
AnswerQuestion 17
Next Question
If using the method of completing the square to solve the quadratic equation x, squared, plus, 8, x, plus, 37, equals, 0x
2
+8x+37=0, which number would have to be added to "complete the square"?
Answer

Answers

Answered by GPT-4o mini
Let's go through the questions one by one.

### Question 1
Rewrite the polynomial in the form \( ax^2 + bx + c \):

The expression given is:
\[
-9 + x^2
\]
Rewriting it gives:
\[
x^2 + 0x - 9
\]
Thus, \( a = 1 \), \( b = 0 \), \( c = -9 \).

### Question 2
Rewrite the polynomial in the form \( ax^2 + bx + c \):

The expression given is:
\[
1 - \frac{1}{6}x^2 - x
\]
Rearranging gives:
\[
-\frac{1}{6}x^2 - x + 1
\]
Thus, \( a = -\frac{1}{6} \), \( b = -1 \), \( c = 1 \).

### Question 3
Rewrite the polynomial in the form \( ax^2 + bx + c \):

The expression given is:
\[
3x + \frac{1}{4}x^2
\]
Rearranging gives:
\[
\frac{1}{4}x^2 + 3x + 0
\]
Thus, \( a = \frac{1}{4} \), \( b = 3 \), \( c = 0 \).

### Question 4
Find the discriminant of the equation \( 6x^2 - 2x - 1 = 0 \):

\[
D = b^2 - 4ac = (-2)^2 - 4(6)(-1) = 4 + 24 = 28
\]

### Question 5
Find the discriminant of the equation \( -2x^2 - 4x + 7 = 0 \):

\[
D = b^2 - 4ac = (-4)^2 - 4(-2)(7) = 16 + 56 = 72
\]

### Question 6
Find the discriminant of the equation \( -9x^2 - x - 5 = 0 \):

\[
D = b^2 - 4ac = (-1)^2 - 4(-9)(-5) = 1 - 180 = -179
\]

### Question 7
Use the quadratic formula to solve \( 8z^2 + 18z - 7 = -2 \):

First, bring it to standard form:
\[
8z^2 + 18z - 5 = 0
\]
Using the quadratic formula:
\[
z = \frac{-b \pm \sqrt{D}}{2a} = \frac{-18 \pm \sqrt{(18)^2 - 4(8)(-5)}}{2(8)}
\]
Calculate \( D \):
\[
D = 18^2 + 160 = 324 + 160 = 484
\]
So,
\[
z = \frac{-18 \pm \sqrt{484}}{16} = \frac{-18 \pm 22}{16}
\]
This gives two solutions:
\[
z = \frac{4}{16} = \frac{1}{4} \quad \text{and} \quad z = \frac{-40}{16} = -\frac{5}{2}
\]

### Question 8
Use the quadratic formula to solve \( 3y^2 - 8y + 4 = 5y \):

First, bring it to standard form:
\[
3y^2 - 13y + 4 = 0
\]
Using the quadratic formula:
\[
y = \frac{-(-13) \pm \sqrt{D}}{2(3)} = \frac{13 \pm \sqrt{(-13)^2 - 4(3)(4)}}{6}
\]
Calculate \( D \):
\[
D = 169 - 48 = 121
\]
So,
\[
y = \frac{13 \pm \sqrt{121}}{6} = \frac{13 \pm 11}{6}
\]
This gives two solutions:
\[
y = \frac{24}{6} = 4 \quad \text{and} \quad y = \frac{2}{6} = \frac{1}{3}
\]

### Question 9
Use the quadratic formula to solve \( 16c^2 - 24c + 4 = -1 \):

First, bring it to standard form:
\[
16c^2 - 24c + 5 = 0
\]
Using the quadratic formula:
\[
c = \frac{-(-24) \pm \sqrt{D}}{2(16)} = \frac{24 \pm \sqrt{D}}{32}
\]
Calculate \( D \):
\[
D = (-24)^2 - 4(16)(5) = 576 - 320 = 256
\]
So,
\[
c = \frac{24 \pm \sqrt{256}}{32} = \frac{24 \pm 16}{32}
\]
This gives two solutions:
\[
c = \frac{40}{32} = \frac{5}{4} \quad \text{and} \quad c = \frac{8}{32} = \frac{1}{4}
\]

### Question 10
Solve the equation \( x^2 + 16x + 56 = 0 \) using the quadratic formula:

Using \( a = 1, b = 16, c = 56 \):
\[
D = b^2 - 4ac = 16^2 - 4(1)(56) = 256 - 224 = 32
\]
So,
\[
x = \frac{-16 \pm \sqrt{32}}{2} = \frac{-16 \pm 4\sqrt{2}}{2} = -8 \pm 2\sqrt{2}
\]
The approximate numerical solutions are:
\[
x \approx -5.17 \quad \text{and} \quad x \approx -10.83
\]

### Question 11
Solve the equation \( 3x^2 - 8x - 14 = 0 \):

Using \( a = 3, b = -8, c = -14 \):
\[
D = (-8)^2 - 4(3)(-14) = 64 + 168 = 232
\]
So,
\[
x = \frac{8 \pm \sqrt{232}}{6} \approx \frac{8 \pm 15.23}{6}
\]
To the nearest tenth:
\[
x \approx 3.7 \quad \text{and} \quad x \approx -1.2
\]

### Question 12
Solve the equation \( 5x^2 - 11x - 7 = 0 \):

Using \( a = 5, b = -11, c = -7 \):
\[
D = (-11)^2 - 4(5)(-7) = 121 + 140 = 261
\]
So,
\[
x = \frac{11 \pm \sqrt{261}}{10} \approx \frac{11 \pm 16.12}{10}
\]
To the nearest tenth:
\[
x \approx 2.7 \quad \text{and} \quad x \approx -0.5
\]

### Question 13
To complete the square for \( x^2 + 3x + 37 = 0 \):

Take the coefficient of \( x \), which is 3, half it to get \( \frac{3}{2} \), and then square it:
\[
\left(\frac{3}{2}\right)^2 = \frac{9}{4}
\]

### Question 14
To complete the square for \( x^2 + 16x - 5 = 0 \):

Take the coefficient of \( x \), which is 16, half it to get \( 8 \), and then square it:
\[
8^2 = 64
\]

### Question 15
To complete the square for \( x^2 + 13x - 3 = 0 \):

Take the coefficient of \( x \), which is 13, half it to get \( \frac{13}{2} \), and then square it:
\[
\left(\frac{13}{2}\right)^2 = \frac{169}{4}
\]

### Question 16
To complete the square for \( x^2 - 19x - 24 = 0 \):

Take the coefficient of \( x \), which is -19, half it to get \( -\frac{19}{2} \), and then square it:
\[
\left(-\frac{19}{2}\right)^2 = \frac{361}{4}
\]

### Question 17
To complete the square for \( x^2 + 8x + 37 = 0 \):

Take the coefficient of \( x \), which is 8, half it to get \( 4 \), and then square it:
\[
4^2 = 16
\]

Feel free to ask for further clarification or more help on any specific question!

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