Solve x^2+2x-24=0 | Microsoft Math Solver (2024)

a+b=2 ab=-24

To solve the equation, factor x^{2}+2x-24 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.

-1,24 -2,12 -3,8 -4,6

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 -24.

-1+24=23 -2+12=10 -3+8=5 -4+6=2

Calculate the sum for each pair.

a=-4 b=6

The solution is the pair that gives sum 2.

\left(x-4\right)\left(x+6\right)

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

x=4 x=-6

To find equation solutions, solve x-4=0 and x+6=0.

a+b=2 ab=1\left(-24\right)=-24

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

-1,24 -2,12 -3,8 -4,6

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 -24.

-1+24=23 -2+12=10 -3+8=5 -4+6=2

Calculate the sum for each pair.

a=-4 b=6

The solution is the pair that gives sum 2.

\left(x^{2}-4x\right)+\left(6x-24\right)

Rewrite x^{2}+2x-24 as \left(x^{2}-4x\right)+\left(6x-24\right).

x\left(x-4\right)+6\left(x-4\right)

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

\left(x-4\right)\left(x+6\right)

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

x=4 x=-6

To find equation solutions, solve x-4=0 and x+6=0.

x^{2}+2x-24=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{-2±\sqrt{2^{2}-4\left(-24\right)}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-2±\sqrt{4-4\left(-24\right)}}{2}

Square 2.

x=\frac{-2±\sqrt{4+96}}{2}

Multiply -4 times -24.

x=\frac{-2±\sqrt{100}}{2}

Add 4 to 96.

x=\frac{-2±10}{2}

Take the square root of 100.

x=\frac{8}{2}

Now solve the equation x=\frac{-2±10}{2} when ± is plus. Add -2 to 10.

x=4

Divide 8 by 2.

x=-\frac{12}{2}

Now solve the equation x=\frac{-2±10}{2} when ± is minus. Subtract 10 from -2.

x=-6

Divide -12 by 2.

x=4 x=-6

The equation is now solved.

x^{2}+2x-24=0

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}+2x-24-\left(-24\right)=-\left(-24\right)

Add 24 to both sides of the equation.

x^{2}+2x=-\left(-24\right)

Subtracting -24 from itself leaves 0.

x^{2}+2x=24

Subtract -24 from 0.

x^{2}+2x+1^{2}=24+1^{2}

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

x^{2}+2x+1=24+1

Square 1.

x^{2}+2x+1=25

Add 24 to 1.

\left(x+1\right)^{2}=25

Factor x^{2}+2x+1. 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+1\right)^{2}}=\sqrt{25}

Take the square root of both sides of the equation.

x+1=5 x+1=-5

Simplify.

x=4 x=-6

Subtract 1 from both sides of the equation.