Solve 2x^2-70x+500

$$2 { x }^{ 2 } -70x+500$$

Factor

$$2\left(x-25\right)\left(x-10\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{2}-35x+250\right)$$

Consider x^{2}-35x+250. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+250. To find a and b, set up a system to be solved.

$$a+b=-35 ab=1\times 250=250$$

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 250.

$$-1,-250 -2,-125 -5,-50 -10,-25$$

Calculate the sum for each pair.

$$-1-250=-251 -2-125=-127 -5-50=-55 -10-25=-35$$

The solution is the pair that gives sum -35.

$$a=-25 b=-10$$

Rewrite x^{2}-35x+250 as \left(x^{2}-25x\right)+\left(-10x+250\right).

$$\left(x^{2}-25x\right)+\left(-10x+250\right)$$

Factor out x in the first and -10 in the second group.

$$x\left(x-25\right)-10\left(x-25\right)$$

Factor out common term x-25 by using distributive property.

$$\left(x-25\right)\left(x-10\right)$$

Rewrite the complete factored expression.

$$2\left(x-25\right)\left(x-10\right)$$

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.

$$2x^{2}-70x+500=0$$

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.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 2\times 500}}{2\times 2}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 2\times 500}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-70\right)±\sqrt{4900-8\times 500}}{2\times 2}$$

Multiply -8 times 500.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4000}}{2\times 2}$$

Add 4900 to -4000.

$$x=\frac{-\left(-70\right)±\sqrt{900}}{2\times 2}$$

Take the square root of 900.

$$x=\frac{-\left(-70\right)±30}{2\times 2}$$

The opposite of -70 is 70.

$$x=\frac{70±30}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{70±30}{4}$$

Now solve the equation x=\frac{70±30}{4} when ± is plus. Add 70 to 30.

$$x=\frac{100}{4}$$

Divide 100 by 4.

$$x=25$$

Now solve the equation x=\frac{70±30}{4} when ± is minus. Subtract 30 from 70.

$$x=\frac{40}{4}$$

Divide 40 by 4.

$$x=10$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 25 for x_{1} and 10 for x_{2}.

$$2x^{2}-70x+500=2\left(x-25\right)\left(x-10\right)$$

Evaluate

$$2\left(x-25\right)\left(x-10\right)$$