Solve for x
x=-3
x=-2
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3x^{2}+15x+18=0
Add 18 to both sides.
x^{2}+5x+6=0
Divide both sides by 3.
a+b=5 ab=1\times 6=6
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+6. To find a and b, set up a system to be solved.
1,6 2,3
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 6.
1+6=7 2+3=5
Calculate the sum for each pair.
a=2 b=3
The solution is the pair that gives sum 5.
\left(x^{2}+2x\right)+\left(3x+6\right)
Rewrite x^{2}+5x+6 as \left(x^{2}+2x\right)+\left(3x+6\right).
x\left(x+2\right)+3\left(x+2\right)
Factor out x in the first and 3 in the second group.
\left(x+2\right)\left(x+3\right)
Factor out common term x+2 by using distributive property.
x=-2 x=-3
To find equation solutions, solve x+2=0 and x+3=0.
3x^{2}+15x=-18
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.
3x^{2}+15x-\left(-18\right)=-18-\left(-18\right)
Add 18 to both sides of the equation.
3x^{2}+15x-\left(-18\right)=0
Subtracting -18 from itself leaves 0.
3x^{2}+15x+18=0
Subtract -18 from 0.
x=\frac{-15±\sqrt{15^{2}-4\times 3\times 18}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 15 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-15±\sqrt{225-4\times 3\times 18}}{2\times 3}
Square 15.
x=\frac{-15±\sqrt{225-12\times 18}}{2\times 3}
Multiply -4 times 3.
x=\frac{-15±\sqrt{225-216}}{2\times 3}
Multiply -12 times 18.
x=\frac{-15±\sqrt{9}}{2\times 3}
Add 225 to -216.
x=\frac{-15±3}{2\times 3}
Take the square root of 9.
x=\frac{-15±3}{6}
Multiply 2 times 3.
x=-\frac{12}{6}
Now solve the equation x=\frac{-15±3}{6} when ± is plus. Add -15 to 3.
x=-2
Divide -12 by 6.
x=-\frac{18}{6}
Now solve the equation x=\frac{-15±3}{6} when ± is minus. Subtract 3 from -15.
x=-3
Divide -18 by 6.
x=-2 x=-3
The equation is now solved.
3x^{2}+15x=-18
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{3x^{2}+15x}{3}=-\frac{18}{3}
Divide both sides by 3.
x^{2}+\frac{15}{3}x=-\frac{18}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+5x=-\frac{18}{3}
Divide 15 by 3.
x^{2}+5x=-6
Divide -18 by 3.
x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-6+\left(\frac{5}{2}\right)^{2}
Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+5x+\frac{25}{4}=-6+\frac{25}{4}
Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+5x+\frac{25}{4}=\frac{1}{4}
Add -6 to \frac{25}{4}.
\left(x+\frac{5}{2}\right)^{2}=\frac{1}{4}
Factor x^{2}+5x+\frac{25}{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{5}{2}\right)^{2}}=\sqrt{\frac{1}{4}}
Take the square root of both sides of the equation.
x+\frac{5}{2}=\frac{1}{2} x+\frac{5}{2}=-\frac{1}{2}
Simplify.
x=-2 x=-3
Subtract \frac{5}{2} from both sides of the equation.
Examples
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{ 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}