Factoring
12x3+11x2+2x
GCF = xx
x(\frac{12{x}^{3}}{x}+\frac{11{x}^{2}}{x}+\frac{2x}{x})x(x12x3+x11x2+x2x)
Simplify each term in parentheses.
x(12{x}^{2}+11x+2)x(12x2+11x+2)
How?
Split the second term in 12{x}^{2}+11x+212x2+11x+2 into two terms.
x(12{x}^{2}+8x+3x+2)x(12x2+8x+3x+2)
Factor out common terms in the first two terms, then in the last two terms.
x(4x(3x+2)+(3x+2))x(4x(3x+2)+(3x+2))
Factor out the common term 3x+23x+2.
x(3x+2)(4x+1)x(3x+2)(4x+1)
{x}^{2}{x}^{3}x2x3
Use Product Rule: {x}^{a}{x}^{b}={x}^{a+b}xaxb=xa+b.
{x}^{2+3}x2+3
2
Simplify 2+32+3 to 55.
{x}^{5}x5
5=x+y2y+x=7
Solve for xx in 5=x+y5=x+y.
x=5-yx=5−y
2
How?
Substitute x=5-yx=5−y into 2y+x=72y+x=7.
y+5=7y+5=7
3
How?
Solve for yy in y+5=7y+5=7.
y=2y=2
4
How?
Substitute y=2y=2 into x=5-yx=5−y.
x=3x=3
5
Therefore,
\begin{aligned}&x=3\\&y=2\end{aligned}x=3y=2
2x+5<7
Subtract 55 from both sides.
2x<7-52x<7−5
2
Simplify 7-57−5 to 22.
2x<22x<2
3
Divide both sides by 22.
x<1x<1
3x2+5x+4
The expression 3{x}^{2}+5x+43x2+5x+4 fits the form a{x}^{2}+bx+cax2+bx+c. Let's complete the square, where:
\begin{aligned}&a=3\\&b=5\\&c=4\end{aligned}a=3b=5c=4
2
Factor out aa, which is 33.
3({x}^{2}+\frac{5}{3}x+\frac{4}{3})3(x2+35x+34)
3
How?
Introduce the constant kk, which is \frac{25}{36}3625 in our case.
3({x}^{2}+\frac{5}{3}x+\frac{25}{36}-\frac{25}{36}+\frac{4}{3})3(x2+35x+3625−3625+34)
4
Use Square of Sum: {(a+b)}^{2}={a}^{2}+2ab+{b}^{2}(a+b)2=a2+2ab+b2.
3({(x+\frac{5}{6})}^{2}-\frac{25}{36}+\frac{4}{3})3((x+65)2−3625+34)
5
Simplify.
3({(x+\frac{5}{6})}^{2}+\frac{23}{36})3((x+65)2+3623)
6
Expand.
3{(x+\frac{5}{6})}^{2}+\frac{23}{12}3(x+65)2+1223
60h2+280h+45
GCF = 55
2
Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
5(\frac{60{h}^{2}}{5}+\frac{280h}{5}+\frac{45}{5})5(560h2+5280h+545)
3
Simplify each term in parentheses.
5(12{h}^{2}+56h+9)5(12h2+56h+9)
4
How?
Split the second term in 12{h}^{2}+56h+912h2+56h+9 into two terms.
5(12{h}^{2}+54h+2h+9)5(12h2+54h+2h+9)
5
Factor out common terms in the first two terms, then in the last two terms.
5(6h(2h+9)+(2h+9))5(6h(2h+9)+(2h+9))
6
Factor out the common term 2h+92h+9.
5(2h+9)(6h+1)5(2h+9)(6h+1)
x3x8
Use Quotient Rule: \frac{{x}^{a}}{{x}^{b}}={x}^{a-b}xbxa=xa−b.
{x}^{8-3}x8−3
2
Simplify 8-38−3 to 55.
{x}^{5}x5
2x−3y=−24x+y=24
Solve for yy in 4x+y=244x+y=24.
y=24-4xy=24−4x
2
How?
Substitute y=24-4xy=24−4x into 2x-3y=-22x−3y=−2.
14x-72=-214x−72=−2
3
How?
Solve for xx in 14x-72=-214x−72=−2.
x=5x=5
4
How?
Substitute x=5x=5 into y=24-4xy=24−4x.
y=4y=4
Therefore,
\begin{aligned}&x=5\\&y=4\end{aligned}x=5y=4
5−x≤6
Subtract 55 from both sides.
-x\le 6-5−x≤6−5
2
Simplify 6-56−5 to 11.
-x\le 1−x≤1
3
Multiply both sides by -1−1.
x\ge -1x≥−1
x2+7x
The expression {x}^{2}+7xx2+7x fits the form a{x}^{2}+bx+cax2+bx+c. Let's complete the square, where:
\begin{aligned}&a=1\\&b=7\\&c=0\end{aligned}a=1b=7c=0
2
How?
Introduce the constant kk, which is \frac{49}{4}449 in our case.
{x}^{2}+7x+\frac{49}{4}-\frac{49}{4}x2+7x+449−449
3
Use Square of Sum: {(a+b)}^{2}={a}^{2}+2ab+{b}^{2}(a+b)2=a2+2ab+b2.
{(x+\frac{7}{2})}^{2}-\frac{49}{4}(x+27)2−449
8{x}^{3}-1258x3−125
Rewrite it in the form {a}^{3}-{b}^{3}a3−b3, where a=2xa=2x and b=5b=5.
{(2x)}^{3}-{5}^{3}(2x)3−53
2
Use Difference of Cubes: {a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2})a3−b3=(a−b)(a2+ab+b2).
(2x-5)({(2x)}^{2}+(2x)(5)+{5}^{2})(2x−5)((2x)2+(2x)(5)+52)
3
Use Multiplication Distributive Property: {(xy)}^{a}={x}^{a}{y}^{a}(xy)a=xaya.
(2x-5)({2}^{2}{x}^{2}+2x\times 5+{5}^{2})(2x−5)(22x2+2x×5+52)
4
Simplify {2}^{2}22 to 44.
(2x-5)(4{x}^{2}+2x\times 5+{5}^{2})(2x−5)(4x2+2x×5+52)
5
Simplify {5}^{2}52 to 2525.
(2x-5)(4{x}^{2}+2x\times 5+25)(2x−5)(4x2+2x×5+25)
6
Simplify 2x\times 52x×5 to 10x10x.
(2x-5)(4{x}^{2}+10x+25)(2x−5)(4x2+10x+25)
{({x}^{8})}^{7}(x8)7
Use Power Rule: {({x}^{a})}^{b}={x}^{ab}(xa)b=xab.
{x}^{56}x56
2x+y=8−6x−3y=10
Solve for yy in 2x+y=82x+y=8.
y=8-2xy=8−2x
2
How?
Substitute y=8-2xy=8−2x into -6x-3y=10−6x−3y=10.
-24=10−24=10
3
Since -24=10−24=10 is not true, this is an inconsistent system.
No Solution
2(x−1)>3(2x+3)
Expand.
2x-2>6x+92x−2>6x+9
2
Subtract 2x2x from both sides.
-2>6x+9-2x−2>6x+9−2x
3
Simplify 6x+9-2x6x+9−2x to 4x+94x+9.
-2>4x+9−2>4x+9
4
Subtract 99 from both sides.
-2-9>4x−2−9>4x
5
Simplify -2-9−2−9 to -11−11.
-11>4x−11>4x
6
Divide both sides by 44.
-\frac{11}{4}>x−411>x
7
Switch sides.
x<-\frac{11}{4}x<−411
3x2+7x
The expression 3{x}^{2}+7x3x2+7x fits the form a{x}^{2}+bx+cax2+bx+c. Let's complete the square, where:
\begin{aligned}&a=3\\&b=7\\&c=0\end{aligned}a=3b=7c=0
2
Factor out aa, which is 33.
3({x}^{2}+\frac{7}{3}x)3(x2+37x)
3
How?
Introduce the constant kk, which is \frac{49}{36}3649 in our case.
3({x}^{2}+\frac{7}{3}x+\frac{49}{36}-\frac{49}{36})3(x2+37x+3649−3649)
4
Use Square of Sum: {(a+b)}^{2}={a}^{2}+2ab+{b}^{2}(a+b)2=a2+2ab+b2.
3({(x+\frac{7}{6})}^{2}-\frac{49}{36})3((x+67)2−3649)
5
Expand.
3{(x+\frac{7}{6})}^{2}-\frac{49}{12}3(x+67)2−1249
-3{x}^{2}+36x-108−3x2+36x−108
Factor out the common term 33.
-3({x}^{2}-12x+36)−3(x2−12x+36)
2
Rewrite {x}^{2}-12x+36x2−12x+36 in the form {a}^{2}-2ab+{b}^{2}a2−2ab+b2, where a=xa=x and b=6b=6.
-3({x}^{2}-2(x)(6)+{6}^{2})−3(x2−2(x)(6)+62)
3
Use Square of Difference: {(a-b)}^{2}={a}^{2}-2ab+{b}^{2}(a−b)2=a2−2ab+b2.
-3{(x-6)}^{2}−3(x−6)2
x−7x9
Use Product Rule: {x}^{a}{x}^{b}={x}^{a+b}xaxb=xa+b.
{x}^{-7+9}x−7+9
2
Simplify -7+9−7+9 to 22.
{x}^{2}x2
5x+y2=710x=8+y2
Solve for xx in 5x+{y}^{2}=75x+y2=7.
x=\frac{7-{y}^{2}}{5}x=57−y2
2
How?
Substitute x=\frac{7-{y}^{2}}{5}x=57−y2 into 10x=8+{y}^{2}10x=8+y2.
2(7-{y}^{2})=8+{y}^{2}2(7−y2)=8+y2
3
How?
Solve for yy in 2(7-{y}^{2})=8+{y}^{2}2(7−y2)=8+y2.
y=\pm \sqrt{2}y=±√2
4
How?
Substitute y=\pm \sqrt{2}y=±√2 into x=\frac{7-{y}^{2}}{5}x=57−y2.
x=1,1x=1,1
5
Therefore,
\begin{aligned}&x=1\\&y=\sqrt{2},-\sqrt{2}\end{aligned}x=1y=√2,−√2
3x+2>5
Subtract 22 from both sides.
3x>5-23x>5−2
2
Simplify 5-25−2 to 33.
3x>33x>3
3
Divide both sides by 33.
x>1x>1
3x3+21x2+36x
Find the Greatest Common Factor (GCF).
GCF = 3x3x
2
Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
3x(\frac{3{x}^{3}}{3x}+\frac{21{x}^{2}}{3x}+\frac{36x}{3x})3x(3x3x3+3x21x2+3x36x)
3
Simplify each term in parentheses.
3x({x}^{2}+7x+12)3x(x2+7x+12)
4
How?
Factor {x}^{2}+7x+12x2+7x+12.
3x(x+3)(x+4)3x(x+3)(x+4)
x^23
Simplify {2}^{3}23 to 88.
{x}^{8}x8
x+y−z=−2z+2x−y=5−x+2y+2z=1
Solve for xx in x+y-z=-2x+y−z=−2.
x=-2-y+zx=−2−y+z
2
How?
Substitute x=-2-y+zx=−2−y+z into z+2x-y=5z+2x−y=5.
3z-4-3y=53z−4−3y=5
3
How?
Substitute x=-2-y+zx=−2−y+z into -x+2y+2z=1−x+2y+2z=1.
2+3y+z=12+3y+z=1
4
How?
Solve for zz in 2+3y+z=12+3y+z=1.
z=-1-3yz=−1−3y
5
How?
Substitute z=-1-3yz=−1−3y into x=-2-y+zx=−2−y+z.
x=-3-4yx=−3−4y
6
How?
Substitute z=-1-3yz=−1−3y into 3z-4-3y=53z−4−3y=5.
-7-12y=5−7−12y=5
7
How?
Solve for yy in -7-12y=5−7−12y=5.
y=-1y=−1
8
How?
Substitute y=-1y=−1 into x=-3-4yx=−3−4y.
x=1x=1
9
How?
Substitute y=-1y=−1 into z=-1-3yz=−1−3y.
z=2z=2
10
Therefore,
\begin{aligned}&x=1\\&y=-1\\&z=2\end{aligned}x=1y=−1z=2
5−3x≤3
Subtract 55 from both sides.
-3x\le 3-5−3x≤3−5
2
Simplify 3-53−5 to -2−2.
-3x\le -2−3x≤−2
3
Divide both sides by -3−3.
x\ge \frac{-2}{-3}x≥−3−2
4
Two negatives make a positive.
x\ge \frac{2}{3}x≥32