Solve x^2-17x+70

$${ x }^{ 2 } -17x+70$$

Factor

$$\left(x-10\right)\left(x-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+70. To find a and b, set up a system to be solved.

$$a+b=-17 ab=1\times 70=70$$

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 70.

$$-1,-70 -2,-35 -5,-14 -7,-10$$

Calculate the sum for each pair.

$$-1-70=-71 -2-35=-37 -5-14=-19 -7-10=-17$$

The solution is the pair that gives sum -17.

$$a=-10 b=-7$$

Rewrite x^{2}-17x+70 as \left(x^{2}-10x\right)+\left(-7x+70\right).

$$\left(x^{2}-10x\right)+\left(-7x+70\right)$$

Factor out x in the first and -7 in the second group.

$$x\left(x-10\right)-7\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(x-7\right)$$

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.

$$x^{2}-17x+70=0$$

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.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 70}}{2}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 70}}{2}$$

Multiply -4 times 70.

$$x=\frac{-\left(-17\right)±\sqrt{289-280}}{2}$$

Add 289 to -280.

$$x=\frac{-\left(-17\right)±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-\left(-17\right)±3}{2}$$

The opposite of -17 is 17.

$$x=\frac{17±3}{2}$$

Now solve the equation x=\frac{17±3}{2} when ± is plus. Add 17 to 3.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{17±3}{2} when ± is minus. Subtract 3 from 17.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and 7 for x_{2}.

$$x^{2}-17x+70=\left(x-10\right)\left(x-7\right)$$

Evaluate

$$\left(x-10\right)\left(x-7\right)$$