Factor

3\left(2-x\right)\left(x+6\right)

$3(2−x)(x+6)$

Solution Steps

Steps Using the Quadratic Formula

Evaluate

3\left(2-x\right)\left(x+6\right)

$3(2−x)(x+6)$

Graph

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3\left(-x^{2}-4x+12\right)

Factor out 3.

a+b=-4 ab=-12=-12

Consider -x^{2}-4x+12. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

1,-12 2,-6 3,-4

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 -12.

1-12=-11 2-6=-4 3-4=-1

Calculate the sum for each pair.

a=2 b=-6

The solution is the pair that gives sum -4.

\left(-x^{2}+2x\right)+\left(-6x+12\right)

Rewrite -x^{2}-4x+12 as \left(-x^{2}+2x\right)+\left(-6x+12\right).

x\left(-x+2\right)+6\left(-x+2\right)

Factor out x in the first and 6 in the second group.

\left(-x+2\right)\left(x+6\right)

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

3\left(-x+2\right)\left(x+6\right)

Rewrite the complete factored expression.

-3x^{2}-12x+36=0

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 36}}{2\left(-3\right)}

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(-12\right)±\sqrt{144-4\left(-3\right)\times 36}}{2\left(-3\right)}

Square -12.

x=\frac{-\left(-12\right)±\sqrt{144+12\times 36}}{2\left(-3\right)}

Multiply -4 times -3.

x=\frac{-\left(-12\right)±\sqrt{144+432}}{2\left(-3\right)}

Multiply 12 times 36.

x=\frac{-\left(-12\right)±\sqrt{576}}{2\left(-3\right)}

Add 144 to 432.

x=\frac{-\left(-12\right)±24}{2\left(-3\right)}

Take the square root of 576.

x=\frac{12±24}{2\left(-3\right)}

The opposite of -12 is 12.

x=\frac{12±24}{-6}

Multiply 2 times -3.

x=\frac{36}{-6}

Now solve the equation x=\frac{12±24}{-6} when ± is plus. Add 12 to 24.

x=-6

Divide 36 by -6.

x=\frac{-12}{-6}

Now solve the equation x=\frac{12±24}{-6} when ± is minus. Subtract 24 from 12.

x=2

Divide -12 by -6.

-3x^{2}-12x+36=-3\left(x-\left(-6\right)\right)\left(x-2\right)

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6 for x_{1} and 2 for x_{2}.

-3x^{2}-12x+36=-3\left(x+6\right)\left(x-2\right)

Simplify all the expressions of the form p-\left(-q\right) to p+q.

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 $