Question

name
math 154b
solving using the quadratic formula worksheet
the quadratic formula:
for quadratic equations: ( ax^2 + bx + c = 0 ),
( x = \frac{ -b pm sqrt{b^2 - 4ac} }{ 2a } )
solve each equation using the quadratic formula.

1. ( 4x^2 + 11x - 20 = 0 )
2. ( x^2 - 5x - 24 = 0 )

Explanation:

Problem 1: Solve \(4x^2 + 11x - 20 = 0\) using the Quadratic Formula

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

For \(ax^2 + bx + c = 0\), here \(a = 4\), \(b = 11\), \(c = -20\).

Step 2: Compute discriminant \(\Delta = b^2 - 4ac\)

\(\Delta = 11^2 - 4(4)(-20) = 121 + 320 = 441\)

Step 3: Apply Quadratic Formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\)

\(x = \frac{-11 \pm \sqrt{441}}{2(4)} = \frac{-11 \pm 21}{8}\)

Step 4: Find two solutions

• For \(+\): \(x = \frac{-11 + 21}{8} = \frac{10}{8} = \frac{5}{4}\)
• For \(-\): \(x = \frac{-11 - 21}{8} = \frac{-32}{8} = -4\)

Problem 2: Solve \(x^2 - 5x - 24 = 0\) using the Quadratic Formula

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

For \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = -5\), \(c = -24\).

Step 2: Compute discriminant \(\Delta = b^2 - 4ac\)

\(\Delta = (-5)^2 - 4(1)(-24) = 25 + 96 = 121\)

Step 3: Apply Quadratic Formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\)

\(x = \frac{-(-5) \pm \sqrt{121}}{2(1)} = \frac{5 \pm 11}{2}\)

Step 4: Find two solutions

• For \(+\): \(x = \frac{5 + 11}{2} = \frac{16}{2} = 8\)
• For \(-\): \(x = \frac{5 - 11}{2} = \frac{-6}{2} = -3\)

Answer:

s:

1. \(x = \frac{5}{4}\) or \(x = -4\)
2. \(x = 8\) or \(x = -3\)