Solve for x

x=-4<br/>x = \frac{9}{2} = 4\frac{1}{2} = 4.5

$x=−4$

$x=29 =421 =4.5$

$x=29 =421 =4.5$

Steps Using Factoring By Grouping

Steps Using the Quadratic Formula

Steps for Completing the Square

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2x^{2}-36-x=0

Subtract x from both sides.

2x^{2}-x-36=0

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

a+b=-1 ab=2\left(-36\right)=-72

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

1,-72 2,-36 3,-24 4,-18 6,-12 8,-9

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -72.

1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1

Calculate the sum for each pair.

a=-9 b=8

The solution is the pair that gives sum -1.

\left(2x^{2}-9x\right)+\left(8x-36\right)

Rewrite 2x^{2}-x-36 as \left(2x^{2}-9x\right)+\left(8x-36\right).

x\left(2x-9\right)+4\left(2x-9\right)

Factor out x in the first and 4 in the second group.

\left(2x-9\right)\left(x+4\right)

Factor out common term 2x-9 by using distributive property.

x=\frac{9}{2} x=-4

To find equation solutions, solve 2x-9=0 and x+4=0.

2x^{2}-36-x=0

Subtract x from both sides.

2x^{2}-x-36=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{-\left(-1\right)±\sqrt{1-4\times 2\left(-36\right)}}{2\times 2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-\left(-1\right)±\sqrt{1-8\left(-36\right)}}{2\times 2}

Multiply -4 times 2.

x=\frac{-\left(-1\right)±\sqrt{1+288}}{2\times 2}

Multiply -8 times -36.

x=\frac{-\left(-1\right)±\sqrt{289}}{2\times 2}

Add 1 to 288.

x=\frac{-\left(-1\right)±17}{2\times 2}

Take the square root of 289.

x=\frac{1±17}{2\times 2}

The opposite of -1 is 1.

x=\frac{1±17}{4}

Multiply 2 times 2.

x=\frac{18}{4}

Now solve the equation x=\frac{1±17}{4} when ± is plus. Add 1 to 17.

x=\frac{9}{2}

Reduce the fraction \frac{18}{4}=4.5 to lowest terms by extracting and canceling out 2.

x=\frac{-16}{4}

Now solve the equation x=\frac{1±17}{4} when ± is minus. Subtract 17 from 1.

x=-4

Divide -16 by 4.

x=\frac{9}{2} x=-4

The equation is now solved.

2x^{2}-36-x=0

Subtract x from both sides.

2x^{2}-x=36

Add 36 to both sides. Anything plus zero gives itself.

\frac{2x^{2}-x}{2}=\frac{36}{2}

Divide both sides by 2.

x^{2}+\frac{-1}{2}x=\frac{36}{2}

Dividing by 2 undoes the multiplication by 2.

x^{2}-\frac{1}{2}x=\frac{36}{2}

Divide -1 by 2.

x^{2}-\frac{1}{2}x=18

Divide 36 by 2.

x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=18+\left(-\frac{1}{4}\right)^{2}

Divide -\frac{1}{2}=-0.5, the coefficient of the x term, by 2 to get -\frac{1}{4}=-0.25. Then add the square of -\frac{1}{4}=-0.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}-\frac{1}{2}x+\frac{1}{16}=18+\frac{1}{16}

Square -\frac{1}{4}=-0.25 by squaring both the numerator and the denominator of the fraction.

x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{289}{16}

Add 18 to \frac{1}{16}=0.0625.

\left(x-\frac{1}{4}\right)^{2}=\frac{289}{16}

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. 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{1}{4}\right)^{2}}=\sqrt{\frac{289}{16}}

Take the square root of both sides of the equation.

x-\frac{1}{4}=\frac{17}{4} x-\frac{1}{4}=-\frac{17}{4}

Simplify.

x=\frac{9}{2} x=-4

Add \frac{1}{4}=0.25 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 $