= 90 + 2ab \Rightarrow 2ab = 54 \Rightarrow ab = 27

= 90 + 2ab \Rightarrow 2ab = 54 \Rightarrow ab = 27

Understanding the Equation: Solving 90 + 2ab = 54 Step-by-Step

In algebra, solving equations step-by-step is essential for clarity and accuracy. One such common problem involves simplifying linear equations with variables—like solving 90 + 2ab = 54 to find the product ab = 27. This article breaks down the process clearly, helping students, tutors, and math learners master basic algebraic manipulation.

What Is the Equation?

We begin with the equation: 90 + 2ab = 54

Our goal is to isolate ab and find its value. This type of equation often appears in math education when teaching simplification and solving for variables—a crucial foundation for more advanced algebra.

Step-by-Step Solution

Step 1: Subtract 90 from both sides To eliminate the constant term on the left, subtract 90 from both sides:

Wait! At this point, we see ab = –18, but this doesn’t match the expected value (ab = 27). This discrepancy indicates a potential typo or misstatement in the original equation.

Fixing the Equation for a Correct Result: ab = 27

Let’s reverse-engineer the problem to achieve ab = 27. Starting again from the original equation:
90 + 2ab = 54, but if we want ab = 27, rearrange the equation correctly: 90 + 2(27) = ? 2ab = 2 × 27 = 54 90 + 54 = 144 ≠ 54 ``

Clearly,90 + 54 = 144—so the original equation90 + 2ab = 54cannot yieldab = 27.

Corrected Equation for ab = 27

To satisfy the conditionab = 27, the original equation must be adjusted. Suppose instead:
What if 90 + 2ab = 90 + 54?
Then:90 + 2ab = 144Subtract 90:2ab = 54Divide by 2:ab = 27✅

Summary: Corrected Workflow

  1. Start with90 + 2ab = 1442. Subtract 90:2ab = 543. Divide by 2:ab = 27

This confirms the logic and shows how precise algebraic manipulation leads to accurate results.

Why Learning This Matters

Understanding how to simplify and solve equations like90 + 2ab = 144is vital for:

  • Building foundational algebra skills
  • Preparing for standardized math tests
  • Supporting problem-solving in science, engineering, and finance

Mistakes in translating real-world problems into equations often happen due to miswritten constants or signs—always double-check values and signs.

Final Thoughts

While90 + 2ab = 54leads toab = –18, correcting constants to match the targetab = 27` reinforces key steps: isolate variables, simplify both sides, and perform inverse operations correctly. Mastering these skills ensures confidence in tackling complex algebraic expressions.

If you’re teaching or learning algebra, visualizing each step and verifying values minimizes errors and deepens understanding—making equations far easier to handle.

Key Takeaways:

  • Always isolate the variable term
  • Perform the same operation on both sides
  • Double-check arithmetic to avoid sign errors
  • Adjust constants to match intended variable values

Related Topics:

  • Solving linear equations for beginners
  • How to simplify algebraic expressions
  • Finding missing factors in equations
  • Applications of ab in real-world problems

Shift your algebra practice today—precision starts with clear, step-by-step reasoning!

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2A^2 + 6Ad + 9d^2 = 16A^2 + 48Ad + 36d^2\Rightarrow 0 = 14A^2 + 42Ad + 27d^2.Divide by $ d^2 $: let $ x = \frac{A}{d} $, then:14x^2 + 42x + 27 = 0.Discriminant: $ 42^2 - 4\cdot14\cdot27 = 1764 - 1512 = 252 = 36 \cdot 7 $.x = \frac{-42 \pm 6\sqrt{7}}{28} = \frac{-21 \pm 3\sqrt{7}}{14}.Third term: $ A + 2d $, first: $ A $, so ratio:\frac{A + 2d}{A} = 1 + 2\frac{d}{A} = 1 + 2\frac{1}{x} = 1 + \frac{2x}{-21 \pm 3\sqrt{7}}.Take positive root for real chance (positive terms): $ x = \frac{-21 + 3\sqrt{7}}{14} $. Compute numerically: $ \sqrt{7} \approx 2.6458 $, $ 3\sqrt{7} \approx 7.937 $, so $ x \approx \frac{-21 + 7.937}{14} = \frac{-13.063}{14} \approx -0.932 $. Then $ \frac{1}{x} \approx -1.073 $, so $ 1 + 2(-1.073) = -1.146 $. Not meaningful.But from quadratic: $ 14x^2 + 42x + 27 = 0 $. Product of roots $ \frac{27}{14} $, sum $ -3 $.