Solve for x
x=3
x = -\frac{5}{2} = -2\frac{1}{2} = -2.5
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a+b=1 ab=-2\times 15=-30
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+15. To find a and b, set up a system to be solved.
-1,30 -2,15 -3,10 -5,6
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 -30.
-1+30=29 -2+15=13 -3+10=7 -5+6=1
Calculate the sum for each pair.
a=6 b=-5
The solution is the pair that gives sum 1.
\left(-2x^{2}+6x\right)+\left(-5x+15\right)
Rewrite -2x^{2}+x+15 as \left(-2x^{2}+6x\right)+\left(-5x+15\right).
2x\left(-x+3\right)+5\left(-x+3\right)
Factor out 2x in the first and 5 in the second group.
\left(-x+3\right)\left(2x+5\right)
Factor out common term -x+3 by using distributive property.
x=3 x=-\frac{5}{2}
To find equation solutions, solve -x+3=0 and 2x+5=0.
-2x^{2}+x+15=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{-1±\sqrt{1^{2}-4\left(-2\right)\times 15}}{2\left(-2\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-2\right)\times 15}}{2\left(-2\right)}
Square 1.
x=\frac{-1±\sqrt{1+8\times 15}}{2\left(-2\right)}
Multiply -4 times -2.
x=\frac{-1±\sqrt{1+120}}{2\left(-2\right)}
Multiply 8 times 15.
x=\frac{-1±\sqrt{121}}{2\left(-2\right)}
Add 1 to 120.
x=\frac{-1±11}{2\left(-2\right)}
Take the square root of 121.
x=\frac{-1±11}{-4}
Multiply 2 times -2.
x=\frac{10}{-4}
Now solve the equation x=\frac{-1±11}{-4} when ± is plus. Add -1 to 11.
x=-\frac{5}{2}
Reduce the fraction \frac{10}{-4} to lowest terms by extracting and canceling out 2.
x=-\frac{12}{-4}
Now solve the equation x=\frac{-1±11}{-4} when ± is minus. Subtract 11 from -1.
x=3
Divide -12 by -4.
x=-\frac{5}{2} x=3
The equation is now solved.
-2x^{2}+x+15=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.
-2x^{2}+x+15-15=-15
Subtract 15 from both sides of the equation.
-2x^{2}+x=-15
Subtracting 15 from itself leaves 0.
\frac{-2x^{2}+x}{-2}=-\frac{15}{-2}
Divide both sides by -2.
x^{2}+\frac{1}{-2}x=-\frac{15}{-2}
Dividing by -2 undoes the multiplication by -2.
x^{2}-\frac{1}{2}x=-\frac{15}{-2}
Divide 1 by -2.
x^{2}-\frac{1}{2}x=\frac{15}{2}
Divide -15 by -2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{15}{2}+\left(-\frac{1}{4}\right)^{2}
Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{15}{2}+\frac{1}{16}
Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{121}{16}
Add \frac{15}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{4}\right)^{2}=\frac{121}{16}
Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. 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}{4}\right)^{2}}=\sqrt{\frac{121}{16}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{11}{4} x-\frac{1}{4}=-\frac{11}{4}
Simplify.
x=3 x=-\frac{5}{2}
Add \frac{1}{4} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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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}