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Steps Using the Quadratic Formula
Steps for Completing the Square
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\left(u-3\right)\left(u+2\right)+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)
Variable u cannot be equal to any of the values 3,4 since division by zero is not defined. Multiply both sides of the equation by \left(u-4\right)\left(u-3\right), the least common multiple of u-4,u-3.
u^{2}-u-6+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)
Use the distributive property to multiply u-3 by u+2 and combine like terms.
u^{2}-u-6+\left(u^{2}-7u+12\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)
Use the distributive property to multiply u-4 by u-3 and combine like terms.
u^{2}-u-6-u^{2}+7u-12=\left(u-4\right)\left(u+1\right)
Use the distributive property to multiply u^{2}-7u+12 by -1.
-u-6+7u-12=\left(u-4\right)\left(u+1\right)
Combine u^{2} and -u^{2} to get 0.
6u-6-12=\left(u-4\right)\left(u+1\right)
Combine -u and 7u to get 6u.
6u-18=\left(u-4\right)\left(u+1\right)
Subtract 12 from -6 to get -18.
6u-18=u^{2}-3u-4
Use the distributive property to multiply u-4 by u+1 and combine like terms.
6u-18-u^{2}=-3u-4
Subtract u^{2} from both sides.
6u-18-u^{2}+3u=-4
Add 3u to both sides.
9u-18-u^{2}=-4
Combine 6u and 3u to get 9u.
9u-18-u^{2}+4=0
Add 4 to both sides.
9u-14-u^{2}=0
Add -18 and 4 to get -14.
-u^{2}+9u-14=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.
u=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
u=\frac{-9±\sqrt{81-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}
Square 9.
u=\frac{-9±\sqrt{81+4\left(-14\right)}}{2\left(-1\right)}
Multiply -4 times -1.
u=\frac{-9±\sqrt{81-56}}{2\left(-1\right)}
Multiply 4 times -14.
u=\frac{-9±\sqrt{25}}{2\left(-1\right)}
Add 81 to -56.
u=\frac{-9±5}{2\left(-1\right)}
Take the square root of 25.
u=\frac{-9±5}{-2}
Multiply 2 times -1.
u=\frac{-4}{-2}
Now solve the equation u=\frac{-9±5}{-2} when ± is plus. Add -9 to 5.
u=2
Divide -4 by -2.
u=\frac{-14}{-2}
Now solve the equation u=\frac{-9±5}{-2} when ± is minus. Subtract 5 from -9.
u=7
Divide -14 by -2.
u=2 u=7
The equation is now solved.
\left(u-3\right)\left(u+2\right)+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)
Variable u cannot be equal to any of the values 3,4 since division by zero is not defined. Multiply both sides of the equation by \left(u-4\right)\left(u-3\right), the least common multiple of u-4,u-3.
u^{2}-u-6+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)
Use the distributive property to multiply u-3 by u+2 and combine like terms.
u^{2}-u-6+\left(u^{2}-7u+12\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)
Use the distributive property to multiply u-4 by u-3 and combine like terms.
u^{2}-u-6-u^{2}+7u-12=\left(u-4\right)\left(u+1\right)
Use the distributive property to multiply u^{2}-7u+12 by -1.
-u-6+7u-12=\left(u-4\right)\left(u+1\right)
Combine u^{2} and -u^{2} to get 0.
6u-6-12=\left(u-4\right)\left(u+1\right)
Combine -u and 7u to get 6u.
6u-18=\left(u-4\right)\left(u+1\right)
Subtract 12 from -6 to get -18.
6u-18=u^{2}-3u-4
Use the distributive property to multiply u-4 by u+1 and combine like terms.
6u-18-u^{2}=-3u-4
Subtract u^{2} from both sides.
6u-18-u^{2}+3u=-4
Add 3u to both sides.
9u-18-u^{2}=-4
Combine 6u and 3u to get 9u.
9u-u^{2}=-4+18
Add 18 to both sides.
9u-u^{2}=14
Add -4 and 18 to get 14.
-u^{2}+9u=14
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{-u^{2}+9u}{-1}=\frac{14}{-1}
Divide both sides by -1.
u^{2}+\frac{9}{-1}u=\frac{14}{-1}
Dividing by -1 undoes the multiplication by -1.
u^{2}-9u=\frac{14}{-1}
Divide 9 by -1.
u^{2}-9u=-14
Divide 14 by -1.
u^{2}-9u+\left(-\frac{9}{2}\right)^{2}=-14+\left(-\frac{9}{2}\right)^{2}
Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}=-4.5. Then add the square of -\frac{9}{2}=-4.5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
u^{2}-9u+\frac{81}{4}=-14+\frac{81}{4}
Square -\frac{9}{2}=-4.5 by squaring both the numerator and the denominator of the fraction.
u^{2}-9u+\frac{81}{4}=\frac{25}{4}
Add -14 to \frac{81}{4}=20.25.
\left(u-\frac{9}{2}\right)^{2}=\frac{25}{4}
Factor u^{2}-9u+\frac{81}{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(u-\frac{9}{2}\right)^{2}}=\sqrt{\frac{25}{4}}
Take the square root of both sides of the equation.
u-\frac{9}{2}=\frac{5}{2} u-\frac{9}{2}=-\frac{5}{2}
Simplify.
u=7 u=2
Add \frac{9}{2}=4.5 to both sides of the equation.