Solve The Equation Round To The Nearest Hundredth

Solving Equations and Rounding tothe Nearest Hundredth

Solving an equation and then rounding the answer to the nearest hundredth is a skill that appears in algebra, calculus, physics, finance, and many everyday problem‑solving situations. Whether you are finding the root of a quadratic, determining the break‑even point in a business model, or calculating a dosage in medicine, the final step often requires you to present the solution with two decimal places. This article walks you through the entire process, explains why rounding matters, and provides clear examples you can follow or adapt to your own problems.

Steps to Solve an Equation and Round to the Nearest Hundredth

Below is a general workflow that works for most algebraic equations. Adjust the specific algebraic manipulations depending on the type of equation you face (linear, quadratic, exponential, logarithmic, etc.).

1. Identify the Equation Type

• Linear: (ax + b = 0)
• Quadratic: (ax^2 + bx + c = 0) - Rational: (\frac{p(x)}{q(x)} = 0) - Exponential/Logarithmic: (a^{x}=b) or (\log_{a}(x)=c)

Knowing the form tells you which solving technique to apply.

2. Isolate the Variable Perform inverse operations to get the variable alone on one side of the equation.

• Add or subtract constants from both sides.
• Multiply or divide both sides by coefficients. - For quadratics, you may need to factor, complete the square, or use the quadratic formula.
• For exponentials, take the logarithm of both sides; for logarithms, exponentiate.

3. Solve for the Exact Value

Obtain an exact expression (often a fraction, radical, or irrational number).

• Example: solving (2x - 5 = 0) gives (x = \frac{5}{2}=2.5). - Example: solving (x^2 - 2 = 0) gives (x = \pm\sqrt{2}).

4. Convert to Decimal (if needed)

If the exact form is not already a decimal, use a calculator or long division to obtain a decimal approximation.

• (\sqrt{2} \approx 1.41421356)
• (\frac{7}{3} \approx 2.33333333)

5. Round to the Nearest Hundredth

Look at the third decimal place (the thousandths digit).

• If it is 5 or greater, increase the hundredths digit by one.
• If it is less than 5, leave the hundredths digit unchanged.
• Drop all digits beyond the hundredths place.

6. State the Final Answer Write the rounded value with exactly two decimal places (e.g., (3.14), (-0.57)). Include the appropriate units if the problem supplies them.

Scientific Explanation of Rounding to the Nearest Hundredth Rounding is a way of reducing the precision of a number while keeping its value close to the original. The hundredth place represents (\frac{1}{100}) of a unit. When we round to this place, we are essentially saying: “The true value lies somewhere between this rounded number and the next possible hundredth, and we choose the one that is closest.”

Mathematically, for a real number (x), the rounded value (r) to the nearest hundredth is:

[ r = \frac{\left\lfloor 100x + 0.5 \right\rfloor}{100} ]

where (\lfloor \cdot \rfloor) denotes the floor function (greatest integer less than or equal to the argument). The addition of (0.5) implements the “round‑half‑up” rule: any fractional part of (100x) that is 0.5 or greater pushes the integer part up by one.

Why Two Decimal Places?

• Measurement Limits: Many instruments (e.g., thermometers, scales) have a precision of ±0.01 units. Reporting more digits would imply false accuracy.
• Communication Clarity: Two decimals are easy to read and compare, especially in tables or financial statements.
• Error Propagation: In subsequent calculations, using overly precise numbers can amplify rounding errors; limiting to a sensible precision keeps error bounds predictable.

Worked Examples

Example 1: Linear Equation

Problem: Solve (4x + 7 = 19) and round to the nearest hundredth.

Solution 1. Subtract 7: (4x = 12)
2. Divide by 4: (x = 3)
3. Decimal form: (3.00)
4. Hundredths place is already exact → (x = 3.00)

Example 2: Quadratic Equation Problem: Solve (x^2 - 5x + 6 = 0) and round the larger root to the nearest hundredth.

Solution

1. Factor: ((x-2)(x-3)=0) → roots (x=2) and (x=3).
2. Larger root: (x=3). 3. Decimal: (3.00) → (x = 3.00)

(If the roots were irrational, we would proceed to decimal conversion and rounding.)

Example 3: Irrational Solution from a Quadratic

Problem: Solve (2x^2 - 3x - 1 = 0) and round both solutions to the nearest hundredth.

Solution
Use the quadratic formula (x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}) with (a=2), (b=-3), (c=-1).

[ x = \frac{3 \pm \sqrt{(-3)^{2}-4(2)(-1)}}{4} = \frac{3 \pm \sqrt{9+8}}{4} = \frac{3 \pm \sqrt{17}}{4} ]

Compute (\sqrt{17} \approx 4.1231056).

• Plus case: (x_1 = \frac{3 + 4.1231056}{4} = \frac{7.1231056}{4} \approx 1.7