2x^2+16x+24-এর সমাধান করুন

$$2 x ^ { 2 } + 16 x + 24$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(x^{2}+8x+12\right)$$

Consider x^{2}+8x+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.

$$a+b=8 ab=1\times 12=12$$

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 12.

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

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 8.

$$a=2 b=6$$

Rewrite x^{2}+8x+12 as \left(x^{2}+2x\right)+\left(6x+12\right).

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

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

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

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

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

Rewrite the complete factored expression.

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

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.

$$2x^{2}+16x+24=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\times 24}}{2\times 2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 2\times 24}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-16±\sqrt{256-8\times 24}}{2\times 2}$$

Multiply -8 times 24.

$$x=\frac{-16±\sqrt{256-192}}{2\times 2}$$

Add 256 to -192.

$$x=\frac{-16±\sqrt{64}}{2\times 2}$$

Take the square root of 64.

$$x=\frac{-16±8}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-16±8}{4}$$

Now solve the equation x=\frac{-16±8}{4} when ± is plus. Add -16 to 8.

$$x=-\frac{8}{4}$$

Divide -8 by 4.

$$x=-2$$

Now solve the equation x=\frac{-16±8}{4} when ± is minus. Subtract 8 from -16.

$$x=-\frac{24}{4}$$

Divide -24 by 4.

$$x=-6$$

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

$$2x^{2}+16x+24=2\left(x-\left(-2\right)\right)\left(x-\left(-6\right)\right)$$

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

$$2x^{2}+16x+24=2\left(x+2\right)\left(x+6\right)$$

Evaluate

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