Question

find each product.

1. \t $(x + 2)^2$ \t 11) \t $(x - 3)^2$ \t 12) \t $(x + 10)^2$
2. \t $(2x + 1)^2$ \t 14) \t $(3x - 4)^2$ \t 15) \t $(5x + 2)^2$
3. \t $(x - 3)(x + 3)$ \t 17) \t $(x - 6)(x + 6)$ \t 18) \t $(2x - 5)(2x + 5)$

Explanation:

Step1: Recall the formula for \((a + b)^2\)

The formula for the square of a binomial \((a + b)^2=a^{2}+2ab + b^{2}\), and for \((a - b)^2=a^{2}-2ab + b^{2}\). Also, the difference of squares formula is \((a - b)(a + b)=a^{2}-b^{2}\). Let's solve each problem one by one.

Problem 10: \((x + 2)^2\)

Step2: Apply the \((a + b)^2\) formula

Here \(a = x\) and \(b = 2\). So \((x+2)^{2}=x^{2}+2\times x\times2 + 2^{2}=x^{2}+4x + 4\)

Problem 11: \((x - 3)^2\)

Step3: Apply the \((a - b)^2\) formula

Here \(a=x\) and \(b = 3\). So \((x - 3)^{2}=x^{2}-2\times x\times3+3^{2}=x^{2}-6x + 9\)

Problem 12: \((x + 10)^2\)

Step4: Apply the \((a + b)^2\) formula

Here \(a = x\) and \(b = 10\). So \((x + 10)^{2}=x^{2}+2\times x\times10+10^{2}=x^{2}+20x + 100\)

Problem 13: \((2x + 1)^2\)

Step5: Apply the \((a + b)^2\) formula

Here \(a = 2x\) and \(b = 1\). So \((2x+1)^{2}=(2x)^{2}+2\times(2x)\times1+1^{2}=4x^{2}+4x + 1\)

Problem 14: \((3x - 4)^2\)

Step6: Apply the \((a - b)^2\) formula

Here \(a = 3x\) and \(b = 4\). So \((3x - 4)^{2}=(3x)^{2}-2\times(3x)\times4 + 4^{2}=9x^{2}-24x + 16\)

Problem 15: \((5x + 2)^2\)

Step7: Apply the \((a + b)^2\) formula

Here \(a=5x\) and \(b = 2\). So \((5x + 2)^{2}=(5x)^{2}+2\times(5x)\times2+2^{2}=25x^{2}+20x + 4\)

Problem 16: \((x - 3)(x + 3)\)

Step8: Apply the difference of squares formula

Here \(a=x\) and \(b = 3\). So \((x - 3)(x + 3)=x^{2}-3^{2}=x^{2}-9\)

Problem 17: \((x - 6)(x + 6)\)

Step9: Apply the difference of squares formula

Here \(a=x\) and \(b = 6\). So \((x - 6)(x + 6)=x^{2}-6^{2}=x^{2}-36\)

Problem 18: \((2x - 5)(2x + 5)\)

Step10: Apply the difference of squares formula

Here \(a = 2x\) and \(b = 5\). So \((2x - 5)(2x + 5)=(2x)^{2}-5^{2}=4x^{2}-25\)

Answer:

1. \(x^{2}+4x + 4\)
2. \(x^{2}-6x + 9\)
3. \(x^{2}+20x + 100\)
4. \(4x^{2}+4x + 1\)
5. \(9x^{2}-24x + 16\)
6. \(25x^{2}+20x + 4\)
7. \(x^{2}-9\)
8. \(x^{2}-36\)
9. \(4x^{2}-25\)