Solve for x
x = -\frac{7}{2} = -3\frac{1}{2} = -3.5
x=6
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a+b=-5 ab=2\left(-42\right)=-84
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-42. To find a and b, set up a system to be solved.
1,-84 2,-42 3,-28 4,-21 6,-14 7,-12
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -84.
1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5
Calculate the sum for each pair.
a=-12 b=7
The solution is the pair that gives sum -5.
\left(2x^{2}-12x\right)+\left(7x-42\right)
Rewrite 2x^{2}-5x-42 as \left(2x^{2}-12x\right)+\left(7x-42\right).
2x\left(x-6\right)+7\left(x-6\right)
Factor out 2x in the first and 7 in the second group.
\left(x-6\right)\left(2x+7\right)
Factor out common term x-6 by using distributive property.
x=6 x=-\frac{7}{2}
To find equation solutions, solve x-6=0 and 2x+7=0.
2x^{2}-5x-42=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{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\left(-42\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -5 for b, and -42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-42\right)}}{2\times 2}
Square -5.
x=\frac{-\left(-5\right)±\sqrt{25-8\left(-42\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-5\right)±\sqrt{25+336}}{2\times 2}
Multiply -8 times -42.
x=\frac{-\left(-5\right)±\sqrt{361}}{2\times 2}
Add 25 to 336.
x=\frac{-\left(-5\right)±19}{2\times 2}
Take the square root of 361.
x=\frac{5±19}{2\times 2}
The opposite of -5 is 5.
x=\frac{5±19}{4}
Multiply 2 times 2.
x=\frac{24}{4}
Now solve the equation x=\frac{5±19}{4} when ± is plus. Add 5 to 19.
x=6
Divide 24 by 4.
x=-\frac{14}{4}
Now solve the equation x=\frac{5±19}{4} when ± is minus. Subtract 19 from 5.
x=-\frac{7}{2}
Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.
x=6 x=-\frac{7}{2}
The equation is now solved.
2x^{2}-5x-42=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}-5x-42-\left(-42\right)=-\left(-42\right)
Add 42 to both sides of the equation.
2x^{2}-5x=-\left(-42\right)
Subtracting -42 from itself leaves 0.
2x^{2}-5x=42
Subtract -42 from 0.
\frac{2x^{2}-5x}{2}=\frac{42}{2}
Divide both sides by 2.
x^{2}-\frac{5}{2}x=\frac{42}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-\frac{5}{2}x=21
Divide 42 by 2.
x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=21+\left(-\frac{5}{4}\right)^{2}
Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{5}{2}x+\frac{25}{16}=21+\frac{25}{16}
Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{361}{16}
Add 21 to \frac{25}{16}.
\left(x-\frac{5}{4}\right)^{2}=\frac{361}{16}
Factor x^{2}-\frac{5}{2}x+\frac{25}{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{5}{4}\right)^{2}}=\sqrt{\frac{361}{16}}
Take the square root of both sides of the equation.
x-\frac{5}{4}=\frac{19}{4} x-\frac{5}{4}=-\frac{19}{4}
Simplify.
x=6 x=-\frac{7}{2}
Add \frac{5}{4} to 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}