3x^2-33x+72=-এর সমাধান করুন

$$3 x ^ { 2 } - 33 x + 72 =$$

Factor

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

Solution Steps

Factor out 3.

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

Consider x^{2}-11x+24. 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 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 -11.

$$a=-8 b=-3$$

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

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

Factor out x in the first and -3 in the second group.

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

Factor out common term x-8 by using distributive property.

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

Rewrite the complete factored expression.

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

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.

$$3x^{2}-33x+72=0$$

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.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 3\times 72}}{2\times 3}$$

Square -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\times 3\times 72}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-33\right)±\sqrt{1089-12\times 72}}{2\times 3}$$

Multiply -12 times 72.

$$x=\frac{-\left(-33\right)±\sqrt{1089-864}}{2\times 3}$$

Add 1089 to -864.

$$x=\frac{-\left(-33\right)±\sqrt{225}}{2\times 3}$$

Take the square root of 225.

$$x=\frac{-\left(-33\right)±15}{2\times 3}$$

The opposite of -33 is 33.

$$x=\frac{33±15}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{33±15}{6}$$

Now solve the equation x=\frac{33±15}{6} when ± is plus. Add 33 to 15.

$$x=\frac{48}{6}$$

Divide 48 by 6.

$$x=8$$

Now solve the equation x=\frac{33±15}{6} when ± is minus. Subtract 15 from 33.

$$x=\frac{18}{6}$$

Divide 18 by 6.

$$x=3$$

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

$$3x^{2}-33x+72=3\left(x-8\right)\left(x-3\right)$$

Evaluate

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