Solve for x
x=-12
x=-3
Graph
Share
Copied to clipboard
a+b=15 ab=36
To solve the equation, factor x^{2}+15x+36 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
1,36 2,18 3,12 4,9 6,6
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 36.
1+36=37 2+18=20 3+12=15 4+9=13 6+6=12
Calculate the sum for each pair.
a=3 b=12
The solution is the pair that gives sum 15.
\left(x+3\right)\left(x+12\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=-3 x=-12
To find equation solutions, solve x+3=0 and x+12=0.
a+b=15 ab=1\times 36=36
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+36. To find a and b, set up a system to be solved.
1,36 2,18 3,12 4,9 6,6
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 36.
1+36=37 2+18=20 3+12=15 4+9=13 6+6=12
Calculate the sum for each pair.
a=3 b=12
The solution is the pair that gives sum 15.
\left(x^{2}+3x\right)+\left(12x+36\right)
Rewrite x^{2}+15x+36 as \left(x^{2}+3x\right)+\left(12x+36\right).
x\left(x+3\right)+12\left(x+3\right)
Factor out x in the first and 12 in the second group.
\left(x+3\right)\left(x+12\right)
Factor out common term x+3 by using distributive property.
x=-3 x=-12
To find equation solutions, solve x+3=0 and x+12=0.
x^{2}+15x+36=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{-15±\sqrt{15^{2}-4\times 36}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 15 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-15±\sqrt{225-4\times 36}}{2}
Square 15.
x=\frac{-15±\sqrt{225-144}}{2}
Multiply -4 times 36.
x=\frac{-15±\sqrt{81}}{2}
Add 225 to -144.
x=\frac{-15±9}{2}
Take the square root of 81.
x=-\frac{6}{2}
Now solve the equation x=\frac{-15±9}{2} when ± is plus. Add -15 to 9.
x=-3
Divide -6 by 2.
x=-\frac{24}{2}
Now solve the equation x=\frac{-15±9}{2} when ± is minus. Subtract 9 from -15.
x=-12
Divide -24 by 2.
x=-3 x=-12
The equation is now solved.
x^{2}+15x+36=0
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.
x^{2}+15x+36-36=-36
Subtract 36 from both sides of the equation.
x^{2}+15x=-36
Subtracting 36 from itself leaves 0.
x^{2}+15x+\left(\frac{15}{2}\right)^{2}=-36+\left(\frac{15}{2}\right)^{2}
Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+15x+\frac{225}{4}=-36+\frac{225}{4}
Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+15x+\frac{225}{4}=\frac{81}{4}
Add -36 to \frac{225}{4}.
\left(x+\frac{15}{2}\right)^{2}=\frac{81}{4}
Factor x^{2}+15x+\frac{225}{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{15}{2}\right)^{2}}=\sqrt{\frac{81}{4}}
Take the square root of both sides of the equation.
x+\frac{15}{2}=\frac{9}{2} x+\frac{15}{2}=-\frac{9}{2}
Simplify.
x=-3 x=-12
Subtract \frac{15}{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}