Question

kuta software - infinite algebra 1
using the quadratic formula
solve each equation with the quadratic formula.

1. ( m^2 - 5m - 14 = 0 )
2. ( b^2 - 4b + 4 = 0 )
3. ( 2m^2 + 2m - 12 = 0 )
4. ( 2x^2 - 3x - 5 = 0 )
5. ( x^2 + 4x + 3 = 0 )
6. ( 2x^2 + 3x - 20 = 0 )
7. ( 4b^2 + 8b + 7 = 4 )

( 2m^2 - 7m - 13 = -10 ) (crossed out)

Explanation:

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

1. \( m^2 - 5m - 14 = 0 \)

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

Here, \( a = 1 \), \( b = -5 \), \( c = -14 \)

Step 2: Substitute into quadratic formula

\( m = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)} = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2} \)

Step 3: Find two solutions

\( m_1 = \frac{5 + 9}{2} = \frac{14}{2} = 7 \)
\( m_2 = \frac{5 - 9}{2} = \frac{-4}{2} = -2 \)

Answer:

\( m = 7 \) or \( m = -2 \)

2. \( b^2 - 4b + 4 = 0 \)

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

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

Step 2: Substitute into quadratic formula

\( b = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(4)}}{2(1)} = \frac{4 \pm \sqrt{16 - 16}}{2} = \frac{4 \pm 0}{2} \)

Step 3: Find solution

\( b = \frac{4}{2} = 2 \) (double root)