Solve for x
x=-5
x=4
Graph
Share
Copied to clipboard
3\left(\left(x+3\right)\left(x-3\right)-4\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Multiply both sides of the equation by 6, the least common multiple of 2,3,6.
3\left(x^{2}-9-4\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.
3\left(x^{2}-13\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Subtract 4 from -9 to get -13.
3x^{2}-39-2\left(x-2\right)=\left(x-2\right)^{2}+1
Use the distributive property to multiply 3 by x^{2}-13.
3x^{2}-39-2x+4=\left(x-2\right)^{2}+1
Use the distributive property to multiply -2 by x-2.
3x^{2}-35-2x=\left(x-2\right)^{2}+1
Add -39 and 4 to get -35.
3x^{2}-35-2x=x^{2}-4x+4+1
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.
3x^{2}-35-2x=x^{2}-4x+5
Add 4 and 1 to get 5.
3x^{2}-35-2x-x^{2}=-4x+5
Subtract x^{2} from both sides.
2x^{2}-35-2x=-4x+5
Combine 3x^{2} and -x^{2} to get 2x^{2}.
2x^{2}-35-2x+4x=5
Add 4x to both sides.
2x^{2}-35+2x=5
Combine -2x and 4x to get 2x.
2x^{2}-35+2x-5=0
Subtract 5 from both sides.
2x^{2}-40+2x=0
Subtract 5 from -35 to get -40.
x^{2}-20+x=0
Divide both sides by 2.
x^{2}+x-20=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=1 ab=1\left(-20\right)=-20
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-20. To find a and b, set up a system to be solved.
-1,20 -2,10 -4,5
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -20.
-1+20=19 -2+10=8 -4+5=1
Calculate the sum for each pair.
a=-4 b=5
The solution is the pair that gives sum 1.
\left(x^{2}-4x\right)+\left(5x-20\right)
Rewrite x^{2}+x-20 as \left(x^{2}-4x\right)+\left(5x-20\right).
x\left(x-4\right)+5\left(x-4\right)
Factor out x in the first and 5 in the second group.
\left(x-4\right)\left(x+5\right)
Factor out common term x-4 by using distributive property.
x=4 x=-5
To find equation solutions, solve x-4=0 and x+5=0.
3\left(\left(x+3\right)\left(x-3\right)-4\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Multiply both sides of the equation by 6, the least common multiple of 2,3,6.
3\left(x^{2}-9-4\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.
3\left(x^{2}-13\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Subtract 4 from -9 to get -13.
3x^{2}-39-2\left(x-2\right)=\left(x-2\right)^{2}+1
Use the distributive property to multiply 3 by x^{2}-13.
3x^{2}-39-2x+4=\left(x-2\right)^{2}+1
Use the distributive property to multiply -2 by x-2.
3x^{2}-35-2x=\left(x-2\right)^{2}+1
Add -39 and 4 to get -35.
3x^{2}-35-2x=x^{2}-4x+4+1
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.
3x^{2}-35-2x=x^{2}-4x+5
Add 4 and 1 to get 5.
3x^{2}-35-2x-x^{2}=-4x+5
Subtract x^{2} from both sides.
2x^{2}-35-2x=-4x+5
Combine 3x^{2} and -x^{2} to get 2x^{2}.
2x^{2}-35-2x+4x=5
Add 4x to both sides.
2x^{2}-35+2x=5
Combine -2x and 4x to get 2x.
2x^{2}-35+2x-5=0
Subtract 5 from both sides.
2x^{2}-40+2x=0
Subtract 5 from -35 to get -40.
2x^{2}+2x-40=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-2±\sqrt{2^{2}-4\times 2\left(-40\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 2 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\times 2\left(-40\right)}}{2\times 2}
Square 2.
x=\frac{-2±\sqrt{4-8\left(-40\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-2±\sqrt{4+320}}{2\times 2}
Multiply -8 times -40.
x=\frac{-2±\sqrt{324}}{2\times 2}
Add 4 to 320.
x=\frac{-2±18}{2\times 2}
Take the square root of 324.
x=\frac{-2±18}{4}
Multiply 2 times 2.
x=\frac{16}{4}
Now solve the equation x=\frac{-2±18}{4} when ± is plus. Add -2 to 18.
x=4
Divide 16 by 4.
x=-\frac{20}{4}
Now solve the equation x=\frac{-2±18}{4} when ± is minus. Subtract 18 from -2.
x=-5
Divide -20 by 4.
x=4 x=-5
The equation is now solved.
3\left(\left(x+3\right)\left(x-3\right)-4\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Multiply both sides of the equation by 6, the least common multiple of 2,3,6.
3\left(x^{2}-9-4\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.
3\left(x^{2}-13\right)-2\left(x-2\right)=\left(x-2\right)^{2}+1
Subtract 4 from -9 to get -13.
3x^{2}-39-2\left(x-2\right)=\left(x-2\right)^{2}+1
Use the distributive property to multiply 3 by x^{2}-13.
3x^{2}-39-2x+4=\left(x-2\right)^{2}+1
Use the distributive property to multiply -2 by x-2.
3x^{2}-35-2x=\left(x-2\right)^{2}+1
Add -39 and 4 to get -35.
3x^{2}-35-2x=x^{2}-4x+4+1
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.
3x^{2}-35-2x=x^{2}-4x+5
Add 4 and 1 to get 5.
3x^{2}-35-2x-x^{2}=-4x+5
Subtract x^{2} from both sides.
2x^{2}-35-2x=-4x+5
Combine 3x^{2} and -x^{2} to get 2x^{2}.
2x^{2}-35-2x+4x=5
Add 4x to both sides.
2x^{2}-35+2x=5
Combine -2x and 4x to get 2x.
2x^{2}+2x=5+35
Add 35 to both sides.
2x^{2}+2x=40
Add 5 and 35 to get 40.
\frac{2x^{2}+2x}{2}=\frac{40}{2}
Divide both sides by 2.
x^{2}+\frac{2}{2}x=\frac{40}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+x=\frac{40}{2}
Divide 2 by 2.
x^{2}+x=20
Divide 40 by 2.
x^{2}+x+\left(\frac{1}{2}\right)^{2}=20+\left(\frac{1}{2}\right)^{2}
Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+x+\frac{1}{4}=20+\frac{1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+x+\frac{1}{4}=\frac{81}{4}
Add 20 to \frac{1}{4}.
\left(x+\frac{1}{2}\right)^{2}=\frac{81}{4}
Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{81}{4}}
Take the square root of both sides of the equation.
x+\frac{1}{2}=\frac{9}{2} x+\frac{1}{2}=-\frac{9}{2}
Simplify.
x=4 x=-5
Subtract \frac{1}{2} from both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}