Solve each quadratic equation. Write your answer as a simplified fraction when applicable. Write the lesser number first.

Question

(1 point)
Quadratic Equation Your Answer
1 (x+3)(x−1)=0
x =
and x =
2 (15x−5)(7+x)=0
x =
and x =
3 (13x−2)(x−34)=0
x =
and x =
4 30=x2+13x
x =
and x =

Answer 1

Let's solve each quadratic equation one by one:

1. Equation: \((x+3)(x−1)=0\)

   To solve for \(x\):

   • Set each factor to zero:
     1. \(x + 3 = 0\) ⟹ \(x = -3\)
     2. \(x - 1 = 0\) ⟹ \(x = 1\)

   Answer: \(x = -3\) and \(x = 1\)

2. Equation: \((15x−5)(7+x)=0\)

   To solve for \(x\):

   • Set each factor to zero:
     1. \(15x - 5 = 0\) ⟹ \(15x = 5\) ⟹ \(x = \frac{5}{15} = \frac{1}{3}\)
     2. \(7 + x = 0\) ⟹ \(x = -7\)

   Answer: \(x = -7\) and \(x = \frac{1}{3}\)

3. Equation: \((13x−2)(x−34)=0\)

   To solve for \(x\):

   • Set each factor to zero:
     1. \(13x - 2 = 0\) ⟹ \(13x = 2\) ⟹ \(x = \frac{2}{13}\)
     2. \(x - 34 = 0\) ⟹ \(x = 34\)

   Answer: \(x = \frac{2}{13}\) and \(x = 34\)

4. Equation: \(30 = x^2 + 13x\)

   Rearrange this to standard form: \[ x^2 + 13x - 30 = 0 \]

   Now, we can solve using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = 13\), and \(c = -30\).

   Calculate the discriminant: \[ b^2 - 4ac = 13^2 - 4(1)(-30) = 169 + 120 = 289 \]

   Now, plug this into the quadratic formula: \[ x = \frac{-13 \pm \sqrt{289}}{2(1)} = \frac{-13 \pm 17}{2} \] Which gives:

   1. \(x = \frac{4}{2} = 2\)
   2. \(x = \frac{-30}{2} = -15\)

   Answer: \(x = -15\) and \(x = 2\)

Here are the final answers summarized:

1. Your Answer: \(x = -3\) and \(x = 1\)
2. Your Answer: \(x = -7\) and \(x = \frac{1}{3}\)
3. Your Answer: \(x = \frac{2}{13}\) and \(x = 34\)
4. Your Answer: \(x = -15\) and \(x = 2\)