Formula for the product of a sum by a difference of a binomial - $(a-b)(a+b)=a^2-b^2$

We continue with the next of the abbreviated multiplication formulas $(a-b)(a+b)=a^2-b^2$. Let's look at some problems to illustrate its applications.

Problem 1 Perform the multiplication $(x+y)(x-y)$.

Solution: Now we apply the formula $(a-b)(a+b)=a^2-b^2$, where $a=x$ and $b=y$, hence $(x-y)(x+y)=x^2-y^2$. 

Problem 2 Perform the multiplication $(3x-4y)(3x+4y)$.

Solution: Apply the formula $(a-b)(a+b)=a^2-b^2$, where $a=3x$ and $b=4y$, hence $(3x-4y)(3x+4y)=(3x)^2-(4y)^2=9x^2-16y^2$.

3 Problem Perform the multiplication $(x^2-z)(x^2+z)$.

Solution: Apply the formula $(a-b)(a+b)=a^2-b^2$, where $a=x^2$ and $b=z$, hence $(x^2-z)(x^2+z)=(x^2)^2-(z)^2=x^4-z^2$. Let us recall the power grading property, i.e. $(a^n)^m=a^{n.m}$.

4 Problem Calculate $17.23$ in a rational way.

Solution: Represent the product $17.23$ in the following way $17.23=(20-3)(20+3)$ and apply the formula $(a-b)(a+b)=a^2-b^2$, so $17.23=(20-3)(20+3)=20^2-3^2=400-9=391$.

5 Problem Simplify the expression $(3x-1)(3x+1)-(x-2)(x+2)$. 

Solution: Thus we get that $(3x-1)(3x+1)-(x-2)(x+2)=(3x)^2-1^2-(x^2-4)=9x^2-1-x^2+4=8x^2+3.$

6 Problem Simplify the expression $(x+2)^2-(x+1)(x-1).$

Solution: Apply the formulas $(a+b)^2=a^2+2ab+b^2$ and $(a-b)(a+b)=a^2-b^2$, hence

$(x+2)^2-(x+1)(x-1)=x^2+2.x.2+2^2-(x^2-1^2)=x^2+4x+4-x^2+1=4x+5.$

7 Problem Prove the identity $(a+b)(a-b)+(b+c)(b-c)+(c+a)(c-a)=0$.

Solution:

Let's denote the left side of the equality by $A$ and the right side by $B$.

$A=(a+b)(a-b)+(b+c)(b-c)+(c+a)(c-a)=a^2-b^2+b^2-c^2+c^2-a^2=0$. We have proved that $A=B$, hence the equality is an identity.

8 Problem Perform the multiplication $(x+y+z)(x+y-z)$.

Solution: We will write the given product in the following way $[(x+y)+z][(x+y)-z]$, now we will apply the formula $(a-b)(a+b)=a^2-b^2$, where $a=x+y$ and $b=z$, hence we get that $[(x+y)+z][(x+y)-z]=(x+y)^2-z^2=x^2+2xy+y^2-z^2$.

9 Problem Simplify the expression $(a+b)(b-a)+a(a-4c)$ and find its numerical value at $a=2$, $b=-5$ and $c=3$.

Solution:

$(a+b)(b-a)+a(a-4c)=b^2-a^2+a^2-4ac=b^2-4ac$. Now substitute in the resulting expression with $a=2$, $b=-5$ and $c=3$. Thus we get that the numerical value of the expression is $(-5)^2-4.2.3=25-24=1$.

10 Problem Find the numerical value of the expression $(2x-3)^2-(x-2)(x+2)-(x-1)(3x-2)$ if $x=\frac{1}{3}$.

Solution: Apply the shortcut multiplication formulas $(a-b)^2=a^2-2ab+b^2$ and $(a-b)(a+b)=a^2-b^2$, and the product $(x-1)(3x-2)$, multiplied by the "each by each" rule, we get that $(2x-3)^2-(x-2)(x+2)-(x-1)(3x-2)=4x^2-12x+9-(x^2-4)-(3x^2-2x-3x+2)$. Now we reveal the parentheses and perform the apparitions $4x^2-12x+9-(x^2-4)-(3x^2-2x-3x+2)=4x^2-12x+9-x^2+4-3x^2+2x+3x-2=-7x+11$. Now substitute in the now simplified expression $x$ with $\frac{1}{3}$, where for the numerical value of the expression we get $-7.\frac{1}{3}+11=-\frac{7}{3}+11=\frac{-7+33}{3}=\frac{26}{3}$.

Homework assignments

1. Perform the grading: 

a) $(1-4x)(1+4x);$ b) $(-5+a)(-5-a);$ c) $(3x^2-4y^2)(3x^2+4y^2);$ d) $(\frac{1}{3}x-y)(\frac{1}{3}x+y);$ e) $(a-b-c)(a-b+c). 

2. Calculate the product by applying the sum by difference formula to two numbers:

a) $98.102;$ b) $47.53;$ c) $11.5.10.5.$

3. Compare the values of the expressions $40^2$ and $38.42$.

4. Simplify the expressions:

a) $(x-8)(x+8)-(x-8)^2;$ b) $(a-5)(5+a)-(1-a)^2;$ c) $(x+y)(x-y)+(x+y)^2-2xy.$

5. Find the normal form of the expression:

a) $(y-2)(y+2)(y^2+4);$ b) $(\frac{a}{7}-\frac{6x}{5})(\frac{a}{7}+\frac{6x}{5}).$ 

6. Calculate the value of the expression by simplifying it beforehand: 

a) $(2x+\frac{1}{3})(2x-\frac{1}{3})-x(4x-3)$, with $x=1;$

b) $5x^2-(3+2x)(2x-3)$, at $x=0.3;$

c) $10y^2-(2+3y)(3y-2)$, at $y=0.2$.

7. Prove the identity $(x+y)(x-y)(x^2+y^2)(x^4+y^4)=x^8-y^8.$

8. Prove the identity $(9a-4)(a+1)+(3a-2)(-2-3a)=5a.$

9. Find the numerical value of the expression $A=(x-3)(x-2)(x+3)-(x+2)(x^2-9)-36$ if $x=3-|-2|.$

10. A student randomly chooses one of the expressions $a^2-b^2$, $(a+b)^2$, $(c-d)^2$, $b^2-a^2$, $a^2+b^2$, $(a-b)(a+b)$, $3a-3b$. What is the chance of choosing an expression that is the difference of the squares of two numbers?

11. Find the values of the expression $Q=q(q-1)+(2+q)(2-q)$ for all $q$ for which $q^{1;2;3;4;5}$.

12. Find the normal polynomial identical to the expression:

a) $(a-b)^2-2(a-b)(a+b)+(a+b)^2$; 

b) $(x^2+2)^2-(x-2)(x+2)(x^2+4)$; 

c) $5(a-2)(a+2)-\frac{1}{2}(8a-6)^2+38$;

d) $(a-1)(a^2+1)(a-1)-(a^2-1)^2$;

e) $2(m-n)^2-2(m+n)^2-4(m+n)(m-n)+8mn$;

f) $(2a-1)(2a+1)-[\frac{1}{2}(4a-3)]^2+(2a-13)(a-\frac{1}{4})$.

13. Is it true that for every $x$, the expression $C=(5-x)^2-(x-3)(x+3)+5(2x-5)$ yields $C>0$?