Calculus Practice — Integrals

Solving with the Quadratic Formula:
The equation: \(1x^2 + (1)x + (-2) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 1, c = -2\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (1)^2 - 4 \cdot (1) \cdot (-2) = 9\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(1) \pm \sqrt{9}}{2 \cdot 1}\)

\(x_1 = -2, x_2 = 1\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (8)x + (15) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 8, c = 15\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (8)^2 - 4 \cdot (1) \cdot (15) = 4\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(8) \pm \sqrt{4}}{2 \cdot 1}\)

\(x_1 = -5, x_2 = -3\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-7)x + (12) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -7, c = 12\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-7)^2 - 4 \cdot (1) \cdot (12) = 1\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-7) \pm \sqrt{1}}{2 \cdot 1}\)

\(x_1 = 3, x_2 = 4\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (0)x + (0) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 0, c = 0\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (0)^2 - 4 \cdot (1) \cdot (0) = 0\)

Step 3: \(\Delta = 0\) → one solution:
\(x = \frac{-b}{2a} = \frac{-(0)}{2 \cdot 1} = 0\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (1)x + (0) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 1, c = 0\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (1)^2 - 4 \cdot (1) \cdot (0) = 1\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(1) \pm \sqrt{1}}{2 \cdot 1}\)

\(x_1 = -1, x_2 = 0\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-2)x + (-15) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -2, c = -15\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-2)^2 - 4 \cdot (1) \cdot (-15) = 64\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-2) \pm \sqrt{64}}{2 \cdot 1}\)

\(x_1 = -3, x_2 = 5\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (0)x + (-1) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 0, c = -1\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (0)^2 - 4 \cdot (1) \cdot (-1) = 4\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(0) \pm \sqrt{4}}{2 \cdot 1}\)

\(x_1 = -1, x_2 = 1\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (3)x + (2) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 3, c = 2\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (3)^2 - 4 \cdot (1) \cdot (2) = 1\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(3) \pm \sqrt{1}}{2 \cdot 1}\)

\(x_1 = -2, x_2 = -1\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-1)x + (-20) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -1, c = -20\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-1)^2 - 4 \cdot (1) \cdot (-20) = 81\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-1) \pm \sqrt{81}}{2 \cdot 1}\)

\(x_1 = -4, x_2 = 5\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-9)x + (20) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -9, c = 20\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-9)^2 - 4 \cdot (1) \cdot (20) = 1\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-9) \pm \sqrt{1}}{2 \cdot 1}\)

\(x_1 = 4, x_2 = 5\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-8)x + (15) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -8, c = 15\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-8)^2 - 4 \cdot (1) \cdot (15) = 4\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-8) \pm \sqrt{4}}{2 \cdot 1}\)

\(x_1 = 3, x_2 = 5\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-2)x + (-3) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -2, c = -3\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-2)^2 - 4 \cdot (1) \cdot (-3) = 16\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-2) \pm \sqrt{16}}{2 \cdot 1}\)

\(x_1 = -1, x_2 = 3\)

Solving with the Quadratic Formula:
The equation: \(3x^2 + (15)x + (18) = 0\)

Step 1: Identify the coefficients:
\(a = 3, b = 15, c = 18\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (15)^2 - 4 \cdot (3) \cdot (18) = 9\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(15) \pm \sqrt{9}}{2 \cdot 3}\)

\(x_1 = -3, x_2 = -2\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (-8)x + (6) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = -8, c = 6\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-8)^2 - 4 \cdot (2) \cdot (6) = 16\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 2}\)

\(x_1 = 1, x_2 = 3\)

Solving with the Quadratic Formula:
The equation: \(3x^2 + (-6)x + (-45) = 0\)

Step 1: Identify the coefficients:
\(a = 3, b = -6, c = -45\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-6)^2 - 4 \cdot (3) \cdot (-45) = 576\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-6) \pm \sqrt{576}}{2 \cdot 3}\)

\(x_1 = -3, x_2 = 5\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (9)x + (20) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 9, c = 20\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (9)^2 - 4 \cdot (1) \cdot (20) = 1\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(9) \pm \sqrt{1}}{2 \cdot 1}\)

\(x_1 = -5, x_2 = -4\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (2)x + (-40) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = 2, c = -40\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (2)^2 - 4 \cdot (2) \cdot (-40) = 324\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(2) \pm \sqrt{324}}{2 \cdot 2}\)

\(x_1 = -5, x_2 = 4\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (-4)x + (2) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = -4, c = 2\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-4)^2 - 4 \cdot (2) \cdot (2) = 0\)

Step 3: \(\Delta = 0\) → one solution:
\(x = \frac{-b}{2a} = \frac{-(-4)}{2 \cdot 2} = 1\)

Solving with the Quadratic Formula:
The equation: \(3x^2 + (3)x + (0) = 0\)

Step 1: Identify the coefficients:
\(a = 3, b = 3, c = 0\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (3)^2 - 4 \cdot (3) \cdot (0) = 9\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(3) \pm \sqrt{9}}{2 \cdot 3}\)

\(x_1 = -1, x_2 = 0\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (16)x + (30) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = 16, c = 30\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (16)^2 - 4 \cdot (2) \cdot (30) = 16\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(16) \pm \sqrt{16}}{2 \cdot 2}\)

\(x_1 = -5, x_2 = -3\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (12)x + (10) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = 12, c = 10\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (12)^2 - 4 \cdot (2) \cdot (10) = 64\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(12) \pm \sqrt{64}}{2 \cdot 2}\)

\(x_1 = -5, x_2 = -1\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-7)x + (10) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -7, c = 10\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-7)^2 - 4 \cdot (1) \cdot (10) = 9\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-7) \pm \sqrt{9}}{2 \cdot 1}\)

\(x_1 = 2, x_2 = 5\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-5)x + (4) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -5, c = 4\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-5)^2 - 4 \cdot (1) \cdot (4) = 9\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-5) \pm \sqrt{9}}{2 \cdot 1}\)

\(x_1 = 1, x_2 = 4\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (-2)x + (-24) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = -2, c = -24\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-2)^2 - 4 \cdot (2) \cdot (-24) = 196\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-2) \pm \sqrt{196}}{2 \cdot 2}\)

\(x_1 = -3, x_2 = 4\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (-6)x + (8) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = -6, c = 8\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-6)^2 - 4 \cdot (1) \cdot (8) = 4\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-6) \pm \sqrt{4}}{2 \cdot 1}\)

\(x_1 = 2, x_2 = 4\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (2)x + (1) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 2, c = 1\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (2)^2 - 4 \cdot (1) \cdot (1) = 0\)

Step 3: \(\Delta = 0\) → one solution:
\(x = \frac{-b}{2a} = \frac{-(2)}{2 \cdot 1} = -1\)

Solving with the Quadratic Formula:
The equation: \(1x^2 + (0)x + (-9) = 0\)

Step 1: Identify the coefficients:
\(a = 1, b = 0, c = -9\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (0)^2 - 4 \cdot (1) \cdot (-9) = 36\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(0) \pm \sqrt{36}}{2 \cdot 1}\)

\(x_1 = -3, x_2 = 3\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (4)x + (-16) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = 4, c = -16\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (4)^2 - 4 \cdot (2) \cdot (-16) = 144\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(4) \pm \sqrt{144}}{2 \cdot 2}\)

\(x_1 = -4, x_2 = 2\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (-2)x + (-12) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = -2, c = -12\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-2)^2 - 4 \cdot (2) \cdot (-12) = 100\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-2) \pm \sqrt{100}}{2 \cdot 2}\)

\(x_1 = -2, x_2 = 3\)

Solving with the Quadratic Formula:
The equation: \(2x^2 + (-4)x + (-6) = 0\)

Step 1: Identify the coefficients:
\(a = 2, b = -4, c = -6\)

Step 2: Calculate the discriminant:
\(\Delta = b^2 - 4ac = (-4)^2 - 4 \cdot (2) \cdot (-6) = 64\)

Step 3: \(\Delta > 0\) → two solutions:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 2}\)

\(x_1 = -1, x_2 = 3\)