Solve for x

x=-5<br/>x=4

$x=−5$

$x=4$

$x=4$

Steps Using Factoring

Steps Using Factoring By Grouping

Steps Using the Quadratic Formula

Steps for Completing the Square

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x^{2}+x-20=0

Subtract 20 from both sides.

a+b=1 ab=-20

To solve the equation, factor x^{2}+x-20 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.

-1,20 -2,10 -4,5

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 -20.

-1+20=19 -2+10=8 -4+5=1

Calculate the sum for each pair.

a=-4 b=5

The solution is the pair that gives sum 1.

\left(x-4\right)\left(x+5\right)

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

x=4 x=-5

To find equation solutions, solve x-4=0 and x+5=0.

x^{2}+x-20=0

Subtract 20 from both sides.

a+b=1 ab=1\left(-20\right)=-20

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-20. To find a and b, set up a system to be solved.

-1,20 -2,10 -4,5

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 -20.

-1+20=19 -2+10=8 -4+5=1

Calculate the sum for each pair.

a=-4 b=5

The solution is the pair that gives sum 1.

\left(x^{2}-4x\right)+\left(5x-20\right)

Rewrite x^{2}+x-20 as \left(x^{2}-4x\right)+\left(5x-20\right).

x\left(x-4\right)+5\left(x-4\right)

Factor out x in the first and 5 in the second group.

\left(x-4\right)\left(x+5\right)

Factor out common term x-4 by using distributive property.

x=4 x=-5

To find equation solutions, solve x-4=0 and x+5=0.

x^{2}+x=20

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}+x-20=20-20

Subtract 20 from both sides of the equation.

x^{2}+x-20=0

Subtracting 20 from itself leaves 0.

x=\frac{-1±\sqrt{1^{2}-4\left(-20\right)}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-1±\sqrt{1-4\left(-20\right)}}{2}

Square 1.

x=\frac{-1±\sqrt{1+80}}{2}

Multiply -4 times -20.

x=\frac{-1±\sqrt{81}}{2}

Add 1 to 80.

x=\frac{-1±9}{2}

Take the square root of 81.

x=\frac{8}{2}

Now solve the equation x=\frac{-1±9}{2} when ± is plus. Add -1 to 9.

x=4

Divide 8 by 2.

x=\frac{-10}{2}

Now solve the equation x=\frac{-1±9}{2} when ± is minus. Subtract 9 from -1.

x=-5

Divide -10 by 2.

x=4 x=-5

The equation is now solved.

x^{2}+x=20

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}+x+\left(\frac{1}{2}\right)^{2}=20+\left(\frac{1}{2}\right)^{2}

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

x^{2}+x+\frac{1}{4}=20+\frac{1}{4}

Square \frac{1}{2}=0.5 by squaring both the numerator and the denominator of the fraction.

x^{2}+x+\frac{1}{4}=\frac{81}{4}

Add 20 to \frac{1}{4}=0.25.

\left(x+\frac{1}{2}\right)^{2}=\frac{81}{4}

Factor x^{2}+x+\frac{1}{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+\frac{1}{2}\right)^{2}}=\sqrt{\frac{81}{4}}

Take the square root of both sides of the equation.

x+\frac{1}{2}=\frac{9}{2} x+\frac{1}{2}=-\frac{9}{2}

Simplify.

x=4 x=-5

Subtract \frac{1}{2}=0.5 from 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 $