Calculate Area: Finding (23x+12)×(20x+7) for a Canvas

Polynomial Multiplication with Binomial Expressions

A painter buys a canvas with the following dimensions:

(23x+12)×(20x+7) (23x+12)\times(20x+7)

How much space to paint does she have?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:03 Pay attention to open parentheses properly
00:07 Each term in parentheses multiply by each term in the second parentheses
00:29 Solve each multiplication and then sum up
00:52 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

A painter buys a canvas with the following dimensions:

(23x+12)×(20x+7) (23x+12)\times(20x+7)

How much space to paint does she have?

2

Step-by-step solution

We calculate the area using the distributive property:

23x×20x+23x×7+12×20x+12×7= 23x\times20x+23x\times7+12\times20x+12\times7=

We solve each of the multiplication exercises:

460x2+161x+240x+84= 460x^2+161x+240x+84=

We join the x coefficients:

460x2+401x+84= 460x^2+401x+84=

3

Final Answer

460x2+401x+84 460x^2+401x+84

Key Points to Remember

Essential concepts to master this topic
  • Distributive Property: Each term in first binomial multiplies each term in second
  • FOIL Method: (23x)(20x) = 460x², then (23x)(7) + (12)(20x) + (12)(7)
  • Check: Count terms: should have x², x, and constant terms ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to multiply all term combinations
    Don't just multiply 23x × 20x = 460x² and stop there! This misses three crucial products and gives an incomplete answer. Always use FOIL or distributive property to multiply every term in the first binomial by every term in the second binomial.

Practice Quiz

Test your knowledge with interactive questions

\( 140-70= \)

FAQ

Everything you need to know about this question

Why do I get four separate products when multiplying two binomials?

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Each term in the first binomial (23x + 12) must multiply each term in the second binomial (20x + 7). That gives you: 23x×20x, 23x×7, 12×20x, and 12×7 = four products total!

How do I remember to get all the products?

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Use FOIL: First terms (23x×20x), Outer terms (23x×7), Inner terms (12×20x), Last terms (12×7). This ensures you don't miss any!

Why do I need to combine 161x + 240x in the middle step?

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Both 161x and 240x are like terms (they both have x to the first power). You must combine them: 161x + 240x = 401x to get your final simplified answer.

What if I get the x² term wrong?

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The x² term comes from multiplying the x terms together: 23x × 20x = 460x². If you get this wrong, double-check that you're multiplying both the numbers (23 × 20 = 460) and the variables (x × x = x²).

How can I check if my final answer is correct?

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Your answer should be a quadratic expression with three terms: an x² term, an x term, and a constant. For this problem: 460x2+401x+84 460x^2 + 401x + 84 has all three parts ✓

What does this represent in the real-world context?

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This represents the total area of the canvas the painter can work on. The expression 460x2+401x+84 460x^2 + 401x + 84 tells us how many square units of space she has to paint.

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