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x^{2}+6x+9-25=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.
x^{2}+6x-16=0
Subtract 25 from 9 to get -16.
a+b=6 ab=-16
To solve the equation, factor x^{2}+6x-16 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.
-1,16 -2,8 -4,4
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 -16.
-1+16=15 -2+8=6 -4+4=0
Calculate the sum for each pair.
a=-2 b=8
The solution is the pair that gives sum 6.
\left(x-2\right)\left(x+8\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=2 x=-8
To find equation solutions, solve x-2=0 and x+8=0.
x^{2}+6x+9-25=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.
x^{2}+6x-16=0
Subtract 25 from 9 to get -16.
a+b=6 ab=1\left(-16\right)=-16
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-16. To find a and b, set up a system to be solved.
-1,16 -2,8 -4,4
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 -16.
-1+16=15 -2+8=6 -4+4=0
Calculate the sum for each pair.
a=-2 b=8
The solution is the pair that gives sum 6.
\left(x^{2}-2x\right)+\left(8x-16\right)
Rewrite x^{2}+6x-16 as \left(x^{2}-2x\right)+\left(8x-16\right).
x\left(x-2\right)+8\left(x-2\right)
Factor out x in the first and 8 in the second group.
\left(x-2\right)\left(x+8\right)
Factor out common term x-2 by using distributive property.
x=2 x=-8
To find equation solutions, solve x-2=0 and x+8=0.
x^{2}+6x+9-25=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.
x^{2}+6x-16=0
Subtract 25 from 9 to get -16.
x=\frac{-6±\sqrt{6^{2}-4\left(-16\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\left(-16\right)}}{2}
Square 6.
x=\frac{-6±\sqrt{36+64}}{2}
Multiply -4 times -16.
x=\frac{-6±\sqrt{100}}{2}
Add 36 to 64.
x=\frac{-6±10}{2}
Take the square root of 100.
x=\frac{4}{2}
Now solve the equation x=\frac{-6±10}{2} when ± is plus. Add -6 to 10.
x=2
Divide 4 by 2.
x=-\frac{16}{2}
Now solve the equation x=\frac{-6±10}{2} when ± is minus. Subtract 10 from -6.
x=-8
Divide -16 by 2.
x=2 x=-8
The equation is now solved.
x^{2}+6x+9-25=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.
x^{2}+6x-16=0
Subtract 25 from 9 to get -16.
x^{2}+6x=16
Add 16 to both sides. Anything plus zero gives itself.
x^{2}+6x+3^{2}=16+3^{2}
Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+6x+9=16+9
Square 3.
x^{2}+6x+9=25
Add 16 to 9.
\left(x+3\right)^{2}=25
Factor x^{2}+6x+9. 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+3\right)^{2}}=\sqrt{25}
Take the square root of both sides of the equation.
x+3=5 x+3=-5
Simplify.
x=2 x=-8
Subtract 3 from both sides of the equation.