Solve for x
x=-5
x=-2
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2x^{2}+14x+20=0
Add 20 to both sides.
x^{2}+7x+10=0
Divide both sides by 2.
a+b=7 ab=1\times 10=10
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+10. To find a and b, set up a system to be solved.
1,10 2,5
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 10.
1+10=11 2+5=7
Calculate the sum for each pair.
a=2 b=5
The solution is the pair that gives sum 7.
\left(x^{2}+2x\right)+\left(5x+10\right)
Rewrite x^{2}+7x+10 as \left(x^{2}+2x\right)+\left(5x+10\right).
x\left(x+2\right)+5\left(x+2\right)
Factor out x in the first and 5 in the second group.
\left(x+2\right)\left(x+5\right)
Factor out common term x+2 by using distributive property.
x=-2 x=-5
To find equation solutions, solve x+2=0 and x+5=0.
2x^{2}+14x=-20
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.
2x^{2}+14x-\left(-20\right)=-20-\left(-20\right)
Add 20 to both sides of the equation.
2x^{2}+14x-\left(-20\right)=0
Subtracting -20 from itself leaves 0.
2x^{2}+14x+20=0
Subtract -20 from 0.
x=\frac{-14±\sqrt{14^{2}-4\times 2\times 20}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 14 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-14±\sqrt{196-4\times 2\times 20}}{2\times 2}
Square 14.
x=\frac{-14±\sqrt{196-8\times 20}}{2\times 2}
Multiply -4 times 2.
x=\frac{-14±\sqrt{196-160}}{2\times 2}
Multiply -8 times 20.
x=\frac{-14±\sqrt{36}}{2\times 2}
Add 196 to -160.
x=\frac{-14±6}{2\times 2}
Take the square root of 36.
x=\frac{-14±6}{4}
Multiply 2 times 2.
x=-\frac{8}{4}
Now solve the equation x=\frac{-14±6}{4} when ± is plus. Add -14 to 6.
x=-2
Divide -8 by 4.
x=-\frac{20}{4}
Now solve the equation x=\frac{-14±6}{4} when ± is minus. Subtract 6 from -14.
x=-5
Divide -20 by 4.
x=-2 x=-5
The equation is now solved.
2x^{2}+14x=-20
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{2x^{2}+14x}{2}=-\frac{20}{2}
Divide both sides by 2.
x^{2}+\frac{14}{2}x=-\frac{20}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+7x=-\frac{20}{2}
Divide 14 by 2.
x^{2}+7x=-10
Divide -20 by 2.
x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-10+\left(\frac{7}{2}\right)^{2}
Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+7x+\frac{49}{4}=-10+\frac{49}{4}
Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+7x+\frac{49}{4}=\frac{9}{4}
Add -10 to \frac{49}{4}.
\left(x+\frac{7}{2}\right)^{2}=\frac{9}{4}
Factor x^{2}+7x+\frac{49}{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{7}{2}\right)^{2}}=\sqrt{\frac{9}{4}}
Take the square root of both sides of the equation.
x+\frac{7}{2}=\frac{3}{2} x+\frac{7}{2}=-\frac{3}{2}
Simplify.
x=-2 x=-5
Subtract \frac{7}{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}