x^2+11x+24-এর সমাধান করুন

$$x^2+11x+24$$

Factor

$$\left(x+3\right)\left(x+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\times 24=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

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

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 11.

$$a=3 b=8$$

Rewrite x^{2}+11x+24 as \left(x^{2}+3x\right)+\left(8x+24\right).

$$\left(x^{2}+3x\right)+\left(8x+24\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x+3\right)+8\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+8\right)$$

Steps Using the Quadratic Formula

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.

$$x^{2}+11x+24=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 24}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 24}}{2}$$

Multiply -4 times 24.

$$x=\frac{-11±\sqrt{121-96}}{2}$$

Add 121 to -96.

$$x=\frac{-11±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-11±5}{2}$$

Now solve the equation x=\frac{-11±5}{2} when ± is plus. Add -11 to 5.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Now solve the equation x=\frac{-11±5}{2} when ± is minus. Subtract 5 from -11.

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -8 for x_{2}.

$$x^{2}+11x+24=\left(x-\left(-3\right)\right)\left(x-\left(-8\right)\right)$$

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

$$x^{2}+11x+24=\left(x+3\right)\left(x+8\right)$$

Evaluate

$$\left(x+3\right)\left(x+8\right)$$