x^2-32x-2448-এর সমাধান করুন

$$x ^ { 2 } - 32 x - 2448$$

Factor

$$\left(x-68\right)\left(x+36\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-2448. To find a and b, set up a system to be solved.

$$a+b=-32 ab=1\left(-2448\right)=-2448$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -2448.

$$1,-2448 2,-1224 3,-816 4,-612 6,-408 8,-306 9,-272 12,-204 16,-153 17,-144 18,-136 24,-102 34,-72 36,-68 48,-51$$

Calculate the sum for each pair.

$$1-2448=-2447 2-1224=-1222 3-816=-813 4-612=-608 6-408=-402 8-306=-298 9-272=-263 12-204=-192 16-153=-137 17-144=-127 18-136=-118 24-102=-78 34-72=-38 36-68=-32 48-51=-3$$

The solution is the pair that gives sum -32.

$$a=-68 b=36$$

Rewrite x^{2}-32x-2448 as \left(x^{2}-68x\right)+\left(36x-2448\right).

$$\left(x^{2}-68x\right)+\left(36x-2448\right)$$

Factor out x in the first and 36 in the second group.

$$x\left(x-68\right)+36\left(x-68\right)$$

Factor out common term x-68 by using distributive property.

$$\left(x-68\right)\left(x+36\right)$$

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.

$$x^{2}-32x-2448=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-2448\right)}}{2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\left(-2448\right)}}{2}$$

Multiply -4 times -2448.

$$x=\frac{-\left(-32\right)±\sqrt{1024+9792}}{2}$$

Add 1024 to 9792.

$$x=\frac{-\left(-32\right)±\sqrt{10816}}{2}$$

Take the square root of 10816.

$$x=\frac{-\left(-32\right)±104}{2}$$

The opposite of -32 is 32.

$$x=\frac{32±104}{2}$$

Now solve the equation x=\frac{32±104}{2} when ± is plus. Add 32 to 104.

$$x=\frac{136}{2}$$

Divide 136 by 2.

$$x=68$$

Now solve the equation x=\frac{32±104}{2} when ± is minus. Subtract 104 from 32.

$$x=-\frac{72}{2}$$

Divide -72 by 2.

$$x=-36$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 68 for x_{1} and -36 for x_{2}.

$$x^{2}-32x-2448=\left(x-68\right)\left(x-\left(-36\right)\right)$$

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

$$x^{2}-32x-2448=\left(x-68\right)\left(x+36\right)$$

Evaluate

$$\left(x-68\right)\left(x+36\right)$$