Science Math: Quadratic Equations, Half-Life, Molarity, and Density Explained

A patient receives a 100 MBq dose of iodine-131 for a thyroid treatment. The half-life is 8.02 days. How much radioactive material remains after 10 days? If you said 50 MBq, you confused one half-life with 10 days. The actual answer is 41.6 MBq. Getting this wrong in a clinical setting means miscalculating a radiation dose.

Science math is not optional. It is the difference between a correct lab result and a dangerous error. Four concepts show up across chemistry, physics, and biology: quadratic equations for modeling motion and equilibrium, half-life for radioactive decay and drug elimination, molarity for solution preparation, and density for material identification. This guide covers all four with worked examples.

Quadratic Equations

A quadratic equation has the form:

ax^2 + bx + c = 0

Where a, b, and c are constants and a is not zero. The solutions are given by the quadratic formula:

x = (-b plus or minus sqrt(b^2 - 4ac)) / 2a

The term b^2 - 4ac is called the discriminant. It tells you how many real solutions exist:

• Positive discriminant: Two real solutions
• Zero discriminant: One real solution (repeated)
• Negative discriminant: No real solutions (two complex solutions)

Example: Projectile Motion

A ball is thrown upward at 20 m/s from a height of 5 meters. When does it hit the ground?

The height equation is: h = -4.9t^2 + 20t + 5 (using g = 9.8 m/s^2)

Set h = 0: -4.9t^2 + 20t + 5 = 0

a = -4.9, b = 20, c = 5

Discriminant = 400 - 4(-4.9)(5) = 400 + 98 = 498

t = (-20 plus or minus sqrt(498)) / (2 x -4.9) = (-20 plus or minus 22.32) / -9.8

t = (-20 + 22.32) / -9.8 = -0.237 (reject, negative time) t = (-20 - 22.32) / -9.8 = 4.32 seconds

The ball hits the ground after 4.32 seconds.

Use our Quadratic Equation Calculator to solve any quadratic equation and see step-by-step solutions.

Exponents and Logarithms

Exponents and logarithms are inverse operations. If y = a^x, then x = log_a(y).

Common Bases

• Base 10 (common logarithm): log(100) = 2 because 10^2 = 100
• Base e (natural logarithm): ln(e^3) = 3 because e^3 = e^3
• Base 2 (binary logarithm): log_2(8) = 3 because 2^3 = 8

Logarithm Rules

• log(a x b) = log(a) + log(b)
• log(a / b) = log(a) - log(b)
• log(a^n) = n x log(a)
• log_a(a) = 1
• log_a(1) = 0

Example: pH Calculation

pH = -log[H+], where [H+] is the hydrogen ion concentration in moles per liter.

If [H+] = 2.5 x 10^-4 M: pH = -log(2.5 x 10^-4) = -(log 2.5 + log 10^-4) = -(0.398 - 4) = -(-3.602) = 3.60

This is an acidic solution (pH below 7).

Use our Exponent Calculator for power calculations and our Log Calculator for logarithm calculations with any base.

Half-Life and Radioactive Decay

Half-life is the time required for half of a radioactive substance to decay. The decay follows first-order kinetics.

The Half-Life Formula

N(t) = N0 x (1/2)^(t / T)

Where:

• N(t) = amount remaining after time t
• N0 = initial amount
• t = elapsed time
• T = half-life

The Decay Constant

The decay constant lambda relates to half-life:

lambda = ln(2) / T = 0.693 / T

The exponential decay form: N(t) = N0 x e^(-lambda x t)

Example: Iodine-131

Iodine-131 has a half-life of 8.02 days. A 100 MBq sample is prepared. How much remains after 10 days?

N(10) = 100 x (1/2)^(10 / 8.02) = 100 x (0.5)^1.247 = 100 x 0.416 = 41.6 MBq

After 10 days, 41.6 MBq remains. The sample has lost 58.4% of its activity.

Common Radioisotopes

Isotope	Half-Life	Application
Carbon-14	5,730 years	Archaeological dating
Iodine-131	8.02 days	Thyroid treatment
Technetium-99m	6.01 hours	Medical imaging
Cobalt-60	5.27 years	Cancer radiation therapy
Uranium-238	4.47 billion years	Rock dating
Americium-241	432 years	Smoke detectors

Carbon-14 Dating

Living organisms maintain a constant ratio of carbon-14 to carbon-12. When an organism dies, carbon-14 decays with a half-life of 5,730 years. By measuring the remaining carbon-14, scientists can estimate age.

Example: An artifact has 30% of its original carbon-14. How old is it?

0.30 = (1/2)^(t / 5730) ln(0.30) = (t / 5730) x ln(0.5) -1.204 = (t / 5730) x (-0.693) t = 1.204 x 5730 / 0.693 = 9,961 years

Use our Half-Life Calculator to calculate remaining amounts, elapsed time, or half-life from any two known values.

Molarity: Solution Concentration

Molarity (M) is the number of moles of solute per liter of solution.

Molarity = moles of solute / liters of solution

Example: Preparing a Solution

You need 500 mL of 0.5 M NaCl solution. How much NaCl do you weigh?

Moles needed = 0.5 M x 0.500 L = 0.25 moles Mass = 0.25 moles x 58.44 g/mol (molar mass of NaCl) = 14.61 grams

Dissolve 14.61 grams of NaCl in water and dilute to 500 mL total volume.

Dilution Formula

M1 x V1 = M2 x V2

Where M1 and V1 are the initial concentration and volume, and M2 and V2 are the final concentration and volume.

Example: You have 2.0 M HCl and need 250 mL of 0.5 M HCl.

2.0 x V1 = 0.5 x 250 V1 = 62.5 mL

Measure 62.5 mL of 2.0 M HCl and dilute with water to 250 mL total.

Use our Molarity Calculator to calculate molarity, moles, or volume for any solution, and our Molecular Weight Calculator to calculate the molar mass of any compound.

Density: Mass per Unit Volume

Density is mass divided by volume. It identifies materials and determines whether objects float or sink.

Density = Mass / Volume

Common Densities

Material	Density (g/cm^3)
Water (4 degrees C)	1.000
Ice	0.917
Aluminum	2.70
Iron	7.87
Gold	19.32
Mercury	13.55
Air (sea level, 20 C)	0.00120
Helium	0.000179

Example: Identifying a Metal

A sample has a mass of 67.5 g and a volume of 25.0 cm^3.

Density = 67.5 / 25.0 = 2.70 g/cm^3

The metal is aluminum.

Buoyancy

An object floats if its density is less than the fluid it is in. Ice (0.917 g/cm^3) floats in water (1.000 g/cm^3) because it is less dense. Iron (7.87 g/cm^3) sinks because it is denser than water. A steel ship floats because its hull shape displaces enough water that the effective density (total mass divided by displaced volume) is below 1.0.

Density with Irregular Shapes

For irregular objects, measure volume by water displacement. Drop the object in a graduated cylinder and measure the water level rise.

Example: A rock has a mass of 45.2 g. Water in a cylinder rises from 30.0 mL to 47.5 mL when the rock is added.

Volume = 47.5 - 30.0 = 17.5 mL = 17.5 cm^3 Density = 45.2 / 17.5 = 2.58 g/cm^3

Use our Density Calculator to calculate density, mass, or volume from any two known values.

Real-World Scenarios

Scenario 1: Drug Elimination

A drug has a half-life of 6 hours. A patient receives a 500 mg dose. How much remains after 24 hours?

N(24) = 500 x (1/2)^(24/6) = 500 x (0.5)^4 = 500 x 0.0625 = 31.25 mg

After 24 hours (4 half-lives), only 6.25% of the drug remains. This is why medications with short half-lives need frequent dosing.

Scenario 2: Lab Solution Preparation

You need to prepare 1.0 L of 0.1 M CuSO4 from a 2.0 M stock solution.

Using dilution formula: M1 x V1 = M2 x V2 2.0 x V1 = 0.1 x 1.0 V1 = 0.05 L = 50 mL

Measure 50 mL of 2.0 M stock and dilute with water to 1.0 L total.

Scenario 3: Bridge Cable Tension

A suspension bridge cable forms a parabolic shape. The equation y = 0.002x^2 describes the cable height (in meters) at horizontal distance x from center. The towers are 200 meters apart. How tall are the towers?

At x = 100 (halfway to tower): y = 0.002 x 100^2 = 0.002 x 10,000 = 20 meters.

The towers are 20 meters tall above the lowest point of the cable.

Common Mistakes

1. Confusing half-life with total decay time. One half-life reduces the sample by 50%, not 100%. After two half-lives, 25% remains, not 0%. The sample never truly reaches zero.

2. Using molarity formula with wrong units. Molarity is moles per liter, not moles per milliliter. Always convert volumes to liters before calculating.

3. Forgetting to convert temperature for density. Density changes with temperature. Water is most dense at 4 degrees C (1.000 g/cm^3). At 20 degrees C it is 0.998 g/cm^3. Always note the temperature when reporting density.

4. Sign errors in quadratic equations. The quadratic formula has a negative sign before b. For 2x^2 + 5x - 3 = 0: x = (-5 plus or minus sqrt(25 + 24)) / 4 = (-5 plus or minus 7) / 4. The solutions are x = 0.5 and x = -3.

5. Mixing up log bases. log(100) = 2 (base 10) but ln(100) = 4.605 (base e). Always verify which logarithm your formula requires. pH uses base 10. Radioactive decay uses base e (natural log).

External Research and Resources

• Purdue University: Half-Life Calculations provides detailed worked examples of radioactive decay problems including carbon-14 dating.
• Chemistry LibreTexts: Kinetics of Radioactive Decay is a comprehensive free textbook covering first-order decay kinetics with practice problems.
• Khan Academy: Chemistry offers free video lessons on molarity, stoichiometry, and solution preparation with worked examples.

People Also Ask

How do you solve a quadratic equation?

Use the quadratic formula: x = (-b plus or minus sqrt(b^2 - 4ac)) / 2a, where the equation is ax^2 + bx + c = 0. The discriminant (b^2 - 4ac) tells you how many solutions exist: positive means two real solutions, zero means one, negative means no real solutions. Use our Quadratic Equation Calculator for instant results.

How do you calculate half-life?

The half-life formula is N(t) = N0 x (1/2)^(t/T), where N0 is the initial amount, t is elapsed time, and T is the half-life. For example, 100 MBq of iodine-131 (half-life 8.02 days) after 10 days: 100 x (0.5)^(10/8.02) = 41.6 MBq. Use our Half-Life Calculator for instant results.

What is molarity and how do you calculate it?

Molarity is moles of solute per liter of solution (M = moles / liters). To prepare 500 mL of 0.5 M NaCl: moles = 0.5 x 0.5 = 0.25 mol, mass = 0.25 x 58.44 = 14.61 g. Dissolve 14.61 g NaCl and dilute to 500 mL. Use our Molarity Calculator for any solution.

How do you calculate density?

Density = Mass / Volume. For a sample with mass 67.5 g and volume 25.0 cm^3: density = 67.5 / 25.0 = 2.70 g/cm^3. This identifies the material as aluminum. For irregular shapes, measure volume by water displacement. Use our Density Calculator for instant results.

Related Calculators

• Quadratic Equation Calculator - Solve any quadratic equation
• Half-Life Calculator - Calculate radioactive decay and remaining amounts
• Molarity Calculator - Calculate solution concentration
• Molecular Weight Calculator - Calculate molar mass of compounds
• Density Calculator - Calculate density, mass, or volume
• Exponent Calculator - Calculate powers and exponents
• Log Calculator - Calculate logarithms with any base
• Scientific Calculator - Advanced mathematical calculations
• Statistics Calculator - Descriptive statistics for datasets