Solve 4(a-7)² = (2a-3)²: Finding the Value of a

Q: Why can't I just take the square root of both sides right away?

While $ \sqrt{(a-7)^2} = |a-7| $, you have a coefficient of 4 on the left side! Taking the square root would give you $ 2|a-7| = |2a-3| $, which creates a more complex absolute value equation.

Q: How do I expand (a-7)² correctly?

Use the pattern $ (a-b)^2 = a^2 - 2ab + b^2 $. So $ (a-7)^2 = a^2 - 2(a)(7) + 7^2 = a^2 - 14a + 49 $. Don't forget the middle term!

Q: How do I convert 187/44 to a mixed number?

Divide: 187 ÷ 44 = 4 remainder 11. So $ \frac{187}{44} = 4\frac{11}{44} = 4\frac{1}{4} $ after simplifying the fraction part.

Q: Could there be two solutions to this equation?

Not in this case! When you expand and simplify, you get a linear equation in a, which has exactly one solution. The original squared terms cancel out completely.

Quadratic Equations with Square Root Method

$4(a-7)^2=(2a-3)^2$

Find a

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find A

00:04 We'll use the short multiplication formulas to open the parentheses

00:38 We'll solve the multiplications and squares

00:44 When there's a multiplication in a square, each factor is squared

00:54 We'll properly open parentheses and multiply by each factor

01:14 We'll reduce what we can

01:28 We'll isolate A

01:57 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$4(a-7)^2=(2a-3)^2$

Find a

Step-by-step solution

To solve this problem, we will follow these steps:

Step 1: Expand both sides of the equation $4(a-7)^2 = (2a-3)^2$ .
Step 2: Simplify the resulting expressions.
Step 3: Solve for $a$ by equating the simplified expressions.

Now, let's work through each step:

Step 1: Expand both sides.
- Left-side expansion: $4(a-7)^2 = 4(a^2 - 14a + 49) = 4a^2 - 56a + 196$ .
- Right-side expansion: $(2a-3)^2 = (2a)^2 - 2 \times 2a \times 3 + 3^2 = 4a^2 - 12a + 9$ .

Step 2: Set the expanded expressions equal to each other:
$4a^2 - 56a + 196 = 4a^2 - 12a + 9$ .

Now, subtract $4a^2$ from both sides to simplify:
$-56a + 196 = -12a + 9$ .

Simplify the equation by bringing all terms involving $a$ to one side and constant terms to the other side:
$-56a + 12a = 9 - 196$ .
This simplifies to $-44a = -187$ .

Step 3: Solve for $a$ by dividing both sides by $-44$ :
$a = \frac{187}{44} = 4\frac{1}{4}$ .

Therefore, the solution to the problem is $a = 4\frac{1}{4}$ .

Final Answer

$4\frac{1}{4}$

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: Apply the binomial formula (a-b)² = a² - 2ab + b²
Technique: Expand 4(a-7)² = 4(a² - 14a + 49) = 4a² - 56a + 196
Check: Substitute a = 4¼: 4(4¼-7)² = 4(-2¾)² = 30.25 = (5.5)² ✓

Common Mistakes

Avoid these frequent errors

Taking square root of both sides immediately
Don't take √4(a-7)² = √(2a-3)² and get 2(a-7) = (2a-3) = wrong equation! The coefficient 4 gets lost and creates incorrect solutions. Always expand both sides completely first, then simplify.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x-y)^2 $$

FAQ

Everything you need to know about this question

Why can't I just take the square root of both sides right away?

While $\sqrt{(a-7)^2} = |a-7|$ , you have a coefficient of 4 on the left side! Taking the square root would give you $2|a-7| = |2a-3|$ , which creates a more complex absolute value equation.

How do I expand (a-7)² correctly?

Use the pattern $(a-b)^2 = a^2 - 2ab + b^2$ . So $(a-7)^2 = a^2 - 2(a)(7) + 7^2 = a^2 - 14a + 49$ . Don't forget the middle term!

What if I get a negative number when solving for a?

Negative solutions are completely valid! Always check your work by substituting back into the original equation. If both sides are equal, your negative answer is correct.

How do I convert 187/44 to a mixed number?

Divide: 187 ÷ 44 = 4 remainder 11. So $\frac{187}{44} = 4\frac{11}{44} = 4\frac{1}{4}$ after simplifying the fraction part.

Could there be two solutions to this equation?

Not in this case! When you expand and simplify, you get a linear equation in a, which has exactly one solution. The original squared terms cancel out completely.

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