Solve Quadratic Equation: 36x² - 144x + 144 = 0 Step-by-Step

Q: How do I know when a quadratic is a perfect square?

Look for the pattern $ a^2 - 2ab + b^2 = (a - b)^2 $. In $ x^2 - 4x + 4 $, we have x² - 2(x)(2) + 2², which equals $ (x - 2)^2 $.

Q: Why does this equation have only one solution instead of two?

When a quadratic factors as $ (x - a)^2 = 0 $, it's called a repeated root. The parabola just touches the x-axis at one point instead of crossing it at two points.

Q: What if I can't see the perfect square pattern right away?

Try the discriminant test: if $ b^2 - 4ac $ equals zero, you have a perfect square. For $ x^2 - 4x + 4 $: $ (-4)^2 - 4(1)(4) = 16 - 16 = 0 $ ✓

Quadratic Equations with Perfect Square Factoring

Solve the equation

$36x^2-144x+144=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Pay attention to the coefficients

00:10 Use the roots formula

00:26 Substitute appropriate values according to the given data and solve for X

00:51 Calculate the products and the square

01:18 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the equation

$36x^2-144x+144=0$

Step-by-step solution

To solve the quadratic equation $36x^2 - 144x + 144 = 0$ , we first examine its structure to determine the best method for solution:

Step 1: Simplify the equation.
Notice that each term in the equation $36x^2 - 144x + 144$ is divisible by 36. Let's simplify it by dividing each term by 36:

$x^2 - 4x + 4 = 0$

Step 2: Factor the simplified equation.
The equation $x^2 - 4x + 4$ can be factored as $(x - 2)^2 = 0$ , since both 2 and -2 added yield -4, and multiplied give 4.

Step 3: Solve for x.
Given $(x - 2)^2 = 0$ , the solution is $x - 2 = 0$ , which results in:

$x = 2$

Therefore, the solution to the equation is $x = 2$ .

This corresponds to the provided correct answer choice $x=2$ .

Final Answer

$x=2$

Key Points to Remember

Essential concepts to master this topic

Simplification: Divide all terms by greatest common factor first
Technique: Factor $x^2 - 4x + 4$ as $(x - 2)^2 = 0$
Check: Substitute $x = 2$ : $36(4) - 144(2) + 144 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to simplify coefficients before solving
Don't try to factor 36x² - 144x + 144 directly = unnecessarily complicated work! Large coefficients make factoring much harder and increase chances of arithmetic errors. Always divide by the GCF (36 in this case) to get the simpler form x² - 4x + 4 = 0 first.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

How do I know when a quadratic is a perfect square?

Look for the pattern $a^2 - 2ab + b^2 = (a - b)^2$ . In $x^2 - 4x + 4$ , we have x² - 2(x)(2) + 2², which equals $(x - 2)^2$ .

Why does this equation have only one solution instead of two?

When a quadratic factors as $(x - a)^2 = 0$ , it's called a repeated root. The parabola just touches the x-axis at one point instead of crossing it at two points.

What's the fastest way to check if I can divide by a common factor?

Look at all the coefficients: 36, -144, and 144. Ask yourself: "What's the largest number that divides all of these?" Try common factors like 4, 12, or 36.

Can I use the quadratic formula instead of factoring?

Yes, but factoring is much faster here! The quadratic formula would give $x = \frac{4 ± \sqrt{16 - 16}}{2} = \frac{4 ± 0}{2} = 2$ , confirming our answer.

What if I can't see the perfect square pattern right away?

Try the discriminant test: if $b^2 - 4ac$ equals zero, you have a perfect square. For $x^2 - 4x + 4$ : $(-4)^2 - 4(1)(4) = 16 - 16 = 0$ ✓

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