x^2-10x+24-এর সমাধান করুন

$$x ^ { 2 } - 10 x + 24$$

Factor

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

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.

$$a+b=-10 ab=1\times 24=24$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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 -10.

$$a=-6 b=-4$$

Rewrite x^{2}-10x+24 as \left(x^{2}-6x\right)+\left(-4x+24\right).

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

Factor out x in the first and -4 in the second group.

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

Factor out common term x-6 by using distributive property.

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

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.

$$x^{2}-10x+24=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 24}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 24}}{2}$$

Multiply -4 times 24.

$$x=\frac{-\left(-10\right)±\sqrt{100-96}}{2}$$

Add 100 to -96.

$$x=\frac{-\left(-10\right)±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-\left(-10\right)±2}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±2}{2}$$

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

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

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

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and 4 for x_{2}.

$$x^{2}-10x+24=\left(x-6\right)\left(x-4\right)$$

Evaluate

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