Solve for x

x=-6

x=2

Graph

Copy

Copied to clipboard

x^{2}+4x=12

Multiply 9 and \frac{4}{3} to get 12.

x^{2}+4x-12=0

Subtract 12 from both sides.

a+b=4 ab=-12

To solve the equation, factor x^{2}+4x-12 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.

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

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

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

x=2 x=-6

To find equation solutions, solve x-2=0 and x+6=0.

x^{2}+4x=12

Multiply 9 and \frac{4}{3} to get 12.

x^{2}+4x-12=0

Subtract 12 from both sides.

a+b=4 ab=1\left(-12\right)=-12

To solve the equation, factor the left hand side by grouping. First, left hand side 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 positive, the positive number has greater absolute value than the negative. 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.

x=2 x=-6

To find equation solutions, solve x-2=0 and x+6=0.

x^{2}+4x=12

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^{2}+4x-12=12-12

Subtract 12 from both sides of the equation.

x^{2}+4x-12=0

Subtracting 12 from itself leaves 0.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-4±\sqrt{16-4\left(-12\right)}}{2}

Square 4.

x=\frac{-4±\sqrt{16+48}}{2}

Multiply -4 times -12.

x=\frac{-4±\sqrt{64}}{2}

Add 16 to 48.

x=\frac{-4±8}{2}

Take the square root of 64.

x=\frac{4}{2}

Now solve the equation x=\frac{-4±8}{2} when ± is plus. Add -4 to 8.

x=2

Divide 4 by 2.

x=\frac{-12}{2}

Now solve the equation x=\frac{-4±8}{2} when ± is minus. Subtract 8 from -4.

x=-6

Divide -12 by 2.

x=2 x=-6

The equation is now solved.

x^{2}+4x=12

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.

x^{2}+4x+2^{2}=12+2^{2}

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}+4x+4=12+4

Square 2.

x^{2}+4x+4=16

Add 12 to 4.

\left(x+2\right)^{2}=16

Factor x^{2}+4x+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(x+2\right)^{2}}=\sqrt{16}

Take the square root of both sides of the equation.

x+2=4 x+2=-4

Simplify.

x=2 x=-6

Subtract 2 from both sides of the equation.

Examples

Quadratic equation

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

Trigonometry

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

Linear equation

y = 3x + 4

Arithmetic

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]

Simultaneous equation

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

Differentiation

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

Integration

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

Limits

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