Question

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complete the following problems on notebook paper.

1. ( x^2 + x - 4 = 0 )
2. ( 2x^2 - 7x + 3 = 0 )
3. ( 8x^2 + 2x - 3 = 0 )
4. ( 2x^2 - 3x = 20 )
5. ( 4x^2 + 4x - 9 = 0 )
6. ( 2x^2 - 36 = x )
7. ( 9x^2 + 4x - 16 = 0 )
8. ( 3x^2 - 3x + 7 = 9x + 3 )

we will finish the assignment on tomorrow.
after you finish, take a picture and turn it in.

Answer:

Let's solve these quadratic equations one by one using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for a quadratic equation \( ax^2 + bx + c = 0 \).

Problem 1: \( x^2 + x - 4 = 0 \)

Step 1: Identify \( a \), \( b \), \( c \)

Here, \( a = 1 \), \( b = 1 \), \( c = -4 \).

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = 1^2 - 4(1)(-4) = 1 + 16 = 17 \)

Step 3: Apply the quadratic formula

\( x = \frac{-1 \pm \sqrt{17}}{2(1)} = \frac{-1 \pm \sqrt{17}}{2} \)
So the solutions are \( x = \frac{-1 + \sqrt{17}}{2} \) and \( x = \frac{-1 - \sqrt{17}}{2} \)

Problem 2: \( 2x^2 - 7x + 3 = 0 \)

Step 1: Identify \( a \), \( b \), \( c \)

Here, \( a = 2 \), \( b = -7 \), \( c = 3 \).

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = (-7)^2 - 4(2)(3) = 49 - 24 = 25 \)

Step 3: Apply the quadratic formula

\( x = \frac{-(-7) \pm \sqrt{25}}{2(2)} = \frac{7 \pm 5}{4} \)
We have two solutions:

• \( x = \frac{7 + 5}{4} = \frac{12}{4} = 3 \)
• \( x = \frac{7 - 5}{4} = \frac{2}{4} = \frac{1}{2} \)

Problem 3: \( 8x^2 + 2x - 3 = 0 \)

Step 1: Identify \( a \), \( b \), \( c \)

Here, \( a = 8 \), \( b = 2 \), \( c = -3 \).

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = 2^2 - 4(8)(-3) = 4 + 96 = 100 \)

Step 3: Apply the quadratic formula

\( x = \frac{-2 \pm \sqrt{100}}{2(8)} = \frac{-2 \pm 10}{16} \)
We have two solutions:

• \( x = \frac{-2 + 10}{16} = \frac{8}{16} = \frac{1}{2} \)
• \( x = \frac{-2 - 10}{16} = \frac{-12}{16} = -\frac{3}{4} \)

Problem 4: \( 2x^2 - 3x = 20 \)

Step 1: Rewrite in standard form

\( 2x^2 - 3x - 20 = 0 \)

Step 2: Identify \( a \), \( b \), \( c \)

Here, \( a = 2 \), \( b = -3 \), \( c = -20 \).

Step 3: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = (-3)^2 - 4(2)(-20) = 9 + 160 = 169 \)

Step 4: Apply the quadratic formula

\( x = \frac{-(-3) \pm \sqrt{169}}{2(2)} = \frac{3 \pm 13}{4} \)
We have two solutions:

• \( x = \frac{3 + 13}{4} = \frac{16}{4} = 4 \)
• \( x = \frac{3 - 13}{4} = \frac{-10}{4} = -\frac{5}{2} \)

Problem 5: \( 4x^2 + 4x - 9 = 0 \)

Step 1: Identify \( a \), \( b \), \( c \)

Here, \( a = 4 \), \( b = 4 \), \( c = -9 \).

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = 4^2 - 4(4)(-9) = 16 + 144 = 160 \)

Step 3: Apply the quadratic formula

\( x = \frac{-4 \pm \sqrt{160}}{2(4)} = \frac{-4 \pm 4\sqrt{10}}{8} = \frac{-1 \pm \sqrt{10}}{2} \)
So the solutions are \( x = \frac{-1 + \sqrt{10}}{2} \) and \( x = \frac{-1 - \sqrt{10}}{2} \)

Problem 6: \( 2x^2 - 36 = x \)

Step 1: Rewrite in standard form

\( 2x^2 - x - 36 = 0 \)

Step 2: Identify \( a \), \( b \), \( c \)

Here, \( a = 2 \), \( b = -1 \), \( c = -36 \).

Step 3: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = (-1)^2 - 4(2)(-36) = 1 + 288 = 289 \)

Step 4: Apply the quadratic formula

\( x = \frac{-(-1) \pm \sqrt{289}}{2(2)} = \frac{1 \pm 17}{4} \)
We have two solutions:

• \( x = \frac{1 + 17}{4} = \frac{18}{4} = \frac{9}{2} \)
• \( x = \frac{1 - 17}{4} = \frac{-16}{4} = -4 \)

Problem 7: \( 9x^2 + 4x - 16 = 0 \)

Step 1: Identify \( a \), \( b \), \( c \)

Here, \( a = 9 \), \( b = 4 \), \( c = -16 \).

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = 4^2 - 4(9)(-16) = 16 + 576 = 592 \)

Step 3: Apply the quadratic formula

\( x = \frac{-4 \pm \sqrt{592}}{2(9)} = \frac{-4 \pm 4\sqrt{37}}{18} = \frac{-2 \pm 2\sqrt{37}}{9} \)
So the solutions are \( x = \frac{-2 + 2\sqrt{37}}{9} \) and \( x = \frac{-2 - 2\sqrt{37}}{9} \)

Problem 8: \( 3x^2 - 3x + 7 = 9x + 3 \)

Step 1: Rewrite in standard form

\( 3x^2 - 3x - 9x + 7 - 3 = 0 \)
\( 3x^2 - 12x + 4 = 0 \)

Step 2: Identify \( a \), \( b \), \( c \)

Here, \( a = 3 \), \( b = -12 \), \( c = 4 \).

Step 3: Calculate the discriminant \( D = b^2 - 4ac \)

\( D = (-12)^2 - 4(3)(4) = 144 - 48 = 96 \)

Step 4: Apply the quadratic formula

\( x = \frac{-(-12) \pm \sqrt{96}}{2(3)} = \frac{12 \pm 4\sqrt{6}}{6} = \frac{6 \pm 2\sqrt{6}}{3} \)
So the solutions are \( x = \frac{6 + 2\sqrt{6}}{3} \) and \( x = \frac{6 - 2\sqrt{6}}{3} \)