Solve 10x^4-55x^3+70x^2

$$10 x ^ { 4 } - 55 x ^ { 3 } + 70 x ^ { 2 }$$

Factor

$$5\left(x-2\right)\left(2x-7\right)x^{2}$$

Solution Steps

Factor out 5.

$$5\left(2x^{4}-11x^{3}+14x^{2}\right)$$

Consider 2x^{4}-11x^{3}+14x^{2}. Factor out x^{2}.

$$x^{2}\left(2x^{2}-11x+14\right)$$

Consider 2x^{2}-11x+14. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+14. To find a and b, set up a system to be solved.

$$a+b=-11 ab=2\times 14=28$$

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 28.

$$-1,-28 -2,-14 -4,-7$$

Calculate the sum for each pair.

$$-1-28=-29 -2-14=-16 -4-7=-11$$

The solution is the pair that gives sum -11.

$$a=-7 b=-4$$

Rewrite 2x^{2}-11x+14 as \left(2x^{2}-7x\right)+\left(-4x+14\right).

$$\left(2x^{2}-7x\right)+\left(-4x+14\right)$$

Factor out x in the first and -2 in the second group.

$$x\left(2x-7\right)-2\left(2x-7\right)$$

Factor out common term 2x-7 by using distributive property.

$$\left(2x-7\right)\left(x-2\right)$$

Rewrite the complete factored expression.

$$5x^{2}\left(2x-7\right)\left(x-2\right)$$

Evaluate

$$5\left(x-2\right)\left(2x-7\right)x^{2}$$