Solve for u

u=2<br/>u=7

$u=2$

$u=7$

$u=7$

Steps Using the Quadratic Formula

Steps for Completing the Square

Still have questions?

Ask a tutor instantly - for free

Share

Copy

Copied to clipboard

\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.

Examples

Quadratic equation

{ x } ^ { 2 } - 4 x - 5 = 0

$x_{2}−4x−5=0$

Trigonometry

4 \sin \theta \cos \theta = 2 \sin \theta

$4sinθcosθ=2sinθ$

Linear equation

y = 3x + 4

$y=3x+4$

Arithmetic

699 * 533

$699∗533$

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]

$[25 34 ][2−1 01 35 ]$

Simultaneous equation

\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.

${8x+2y=467x+3y=47 $

Differentiation

\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }

$dxd (x−5)(3x_{2}−2) $

Integration

\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x

$∫_{0}xe_{−x_{2}}dx$

Limits

\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}

$x→−3lim x_{2}+2x−3x_{2}−9 $