Solve for x
x=-15
x=8
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x^{2}+7x-78-42=0
Subtract 42 from both sides.
x^{2}+7x-120=0
Subtract 42 from -78 to get -120.
a+b=7 ab=-120
To solve the equation, factor x^{2}+7x-120 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.
-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12
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 -120.
-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2
Calculate the sum for each pair.
a=-8 b=15
The solution is the pair that gives sum 7.
\left(x-8\right)\left(x+15\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=8 x=-15
To find equation solutions, solve x-8=0 and x+15=0.
x^{2}+7x-78-42=0
Subtract 42 from both sides.
x^{2}+7x-120=0
Subtract 42 from -78 to get -120.
a+b=7 ab=1\left(-120\right)=-120
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-120. To find a and b, set up a system to be solved.
-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12
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 -120.
-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2
Calculate the sum for each pair.
a=-8 b=15
The solution is the pair that gives sum 7.
\left(x^{2}-8x\right)+\left(15x-120\right)
Rewrite x^{2}+7x-120 as \left(x^{2}-8x\right)+\left(15x-120\right).
x\left(x-8\right)+15\left(x-8\right)
Factor out x in the first and 15 in the second group.
\left(x-8\right)\left(x+15\right)
Factor out common term x-8 by using distributive property.
x=8 x=-15
To find equation solutions, solve x-8=0 and x+15=0.
x^{2}+7x-78=42
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^{2}+7x-78-42=42-42
Subtract 42 from both sides of the equation.
x^{2}+7x-78-42=0
Subtracting 42 from itself leaves 0.
x^{2}+7x-120=0
Subtract 42 from -78.
x=\frac{-7±\sqrt{7^{2}-4\left(-120\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±\sqrt{49-4\left(-120\right)}}{2}
Square 7.
x=\frac{-7±\sqrt{49+480}}{2}
Multiply -4 times -120.
x=\frac{-7±\sqrt{529}}{2}
Add 49 to 480.
x=\frac{-7±23}{2}
Take the square root of 529.
x=\frac{16}{2}
Now solve the equation x=\frac{-7±23}{2} when ± is plus. Add -7 to 23.
x=8
Divide 16 by 2.
x=-\frac{30}{2}
Now solve the equation x=\frac{-7±23}{2} when ± is minus. Subtract 23 from -7.
x=-15
Divide -30 by 2.
x=8 x=-15
The equation is now solved.
x^{2}+7x-78=42
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}+7x-78-\left(-78\right)=42-\left(-78\right)
Add 78 to both sides of the equation.
x^{2}+7x=42-\left(-78\right)
Subtracting -78 from itself leaves 0.
x^{2}+7x=120
Subtract -78 from 42.
x^{2}+7x+\left(\frac{7}{2}\right)^{2}=120+\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}=120+\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{529}{4}
Add 120 to \frac{49}{4}.
\left(x+\frac{7}{2}\right)^{2}=\frac{529}{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{529}{4}}
Take the square root of both sides of the equation.
x+\frac{7}{2}=\frac{23}{2} x+\frac{7}{2}=-\frac{23}{2}
Simplify.
x=8 x=-15
Subtract \frac{7}{2} from both sides of the equation.
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