মূল বিষয়বস্তুতে এড়িয়ে যান
$3 \exponential{x}{2} - 10 x + 8 $
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a+b=-10 ab=3\times 8=24
Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+8. To find a and b, set up a system to be solved.
-1,-24 -2,-12 -3,-8 -4,-6
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=-25 -2-12=-14 -3-8=-11 -4-6=-10
Calculate the sum for each pair.
a=-6 b=-4
The solution is the pair that gives sum -10.
\left(3x^{2}-6x\right)+\left(-4x+8\right)
Rewrite 3x^{2}-10x+8 as \left(3x^{2}-6x\right)+\left(-4x+8\right).
3x\left(x-2\right)-4\left(x-2\right)
Factor out 3x in the first and -4 in the second group.
\left(x-2\right)\left(3x-4\right)
Factor out common term x-2 by using distributive property.
3x^{2}-10x+8=0
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=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 3\times 8}}{2\times 3}
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{100-4\times 3\times 8}}{2\times 3}
Square -10.
x=\frac{-\left(-10\right)±\sqrt{100-12\times 8}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-10\right)±\sqrt{100-96}}{2\times 3}
Multiply -12 times 8.
x=\frac{-\left(-10\right)±\sqrt{4}}{2\times 3}
Add 100 to -96.
x=\frac{-\left(-10\right)±2}{2\times 3}
Take the square root of 4.
x=\frac{10±2}{2\times 3}
The opposite of -10 is 10.
x=\frac{10±2}{6}
Multiply 2 times 3.
x=\frac{12}{6}
Now solve the equation x=\frac{10±2}{6} when ± is plus. Add 10 to 2.
x=2
Divide 12 by 6.
x=\frac{8}{6}
Now solve the equation x=\frac{10±2}{6} when ± is minus. Subtract 2 from 10.
x=\frac{4}{3}
Reduce the fraction \frac{8}{6} to lowest terms by extracting and canceling out 2.
3x^{2}-10x+8=3\left(x-2\right)\left(x-\frac{4}{3}\right)
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 \frac{4}{3} for x_{2}.
3x^{2}-10x+8=3\left(x-2\right)\times \frac{3x-4}{3}
Subtract \frac{4}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
3x^{2}-10x+8=\left(x-2\right)\left(3x-4\right)
Cancel out 3, the greatest common factor in 3 and 3.