Solve for x
x=-6
x=4
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Quadratic Equation5 problems similar to: { x }^{ 2 } +2x-24 = 0Similar Problems from Web Search
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a+b=2 ab=-24
To solve the equation, factor x^{2}+2x-24 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.
-1,24 -2,12 -3,8 -4,6
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 -24.
-1+24=23 -2+12=10 -3+8=5 -4+6=2
Calculate the sum for each pair.
a=-4 b=6
The solution is the pair that gives sum 2.
\left(x-4\right)\left(x+6\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=4 x=-6
To find equation solutions, solve x-4=0 and x+6=0.
a+b=2 ab=1\left(-24\right)=-24
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.
-1,24 -2,12 -3,8 -4,6
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 -24.
-1+24=23 -2+12=10 -3+8=5 -4+6=2
Calculate the sum for each pair.
a=-4 b=6
The solution is the pair that gives sum 2.
\left(x^{2}-4x\right)+\left(6x-24\right)
Rewrite x^{2}+2x-24 as \left(x^{2}-4x\right)+\left(6x-24\right).
x\left(x-4\right)+6\left(x-4\right)
Factor out x in the first and 6 in the second group.
\left(x-4\right)\left(x+6\right)
Factor out common term x-4 by using distributive property.
x=4 x=-6
To find equation solutions, solve x-4=0 and x+6=0.
x^{2}+2x-24=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{-2±\sqrt{2^{2}-4\left(-24\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-24\right)}}{2}
Square 2.
x=\frac{-2±\sqrt{4+96}}{2}
Multiply -4 times -24.
x=\frac{-2±\sqrt{100}}{2}
Add 4 to 96.
x=\frac{-2±10}{2}
Take the square root of 100.
x=\frac{8}{2}
Now solve the equation x=\frac{-2±10}{2} when ± is plus. Add -2 to 10.
x=4
Divide 8 by 2.
x=-\frac{12}{2}
Now solve the equation x=\frac{-2±10}{2} when ± is minus. Subtract 10 from -2.
x=-6
Divide -12 by 2.
x=4 x=-6
The equation is now solved.
x^{2}+2x-24=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}+2x-24-\left(-24\right)=-\left(-24\right)
Add 24 to both sides of the equation.
x^{2}+2x=-\left(-24\right)
Subtracting -24 from itself leaves 0.
x^{2}+2x=24
Subtract -24 from 0.
x^{2}+2x+1^{2}=24+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=24+1
Square 1.
x^{2}+2x+1=25
Add 24 to 1.
\left(x+1\right)^{2}=25
Factor x^{2}+2x+1. 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+1\right)^{2}}=\sqrt{25}
Take the square root of both sides of the equation.
x+1=5 x+1=-5
Simplify.
x=4 x=-6
Subtract 1 from both sides of the equation.