Solve 7x^2-37x+10

$$7 { x }^{ 2 } -37x+10$$

Factor

$$\left(x-5\right)\left(7x-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=-37 ab=7\times 10=70$$

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 -37.

$$a=-35 b=-2$$

Rewrite 7x^{2}-37x+10 as \left(7x^{2}-35x\right)+\left(-2x+10\right).

$$\left(7x^{2}-35x\right)+\left(-2x+10\right)$$

Factor out 7x in the first and -2 in the second group.

$$7x\left(x-5\right)-2\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(7x-2\right)$$

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.

$$7x^{2}-37x+10=0$$

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.

$$x=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 7\times 10}}{2\times 7}$$

Square -37.

$$x=\frac{-\left(-37\right)±\sqrt{1369-4\times 7\times 10}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-37\right)±\sqrt{1369-28\times 10}}{2\times 7}$$

Multiply -28 times 10.

$$x=\frac{-\left(-37\right)±\sqrt{1369-280}}{2\times 7}$$

Add 1369 to -280.

$$x=\frac{-\left(-37\right)±\sqrt{1089}}{2\times 7}$$

Take the square root of 1089.

$$x=\frac{-\left(-37\right)±33}{2\times 7}$$

The opposite of -37 is 37.

$$x=\frac{37±33}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{37±33}{14}$$

Now solve the equation x=\frac{37±33}{14} when ± is plus. Add 37 to 33.

$$x=\frac{70}{14}$$

Divide 70 by 14.

$$x=5$$

Now solve the equation x=\frac{37±33}{14} when ± is minus. Subtract 33 from 37.

$$x=\frac{4}{14}$$

Reduce the fraction \frac{4}{14} to lowest terms by extracting and canceling out 2.

$$x=\frac{2}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and \frac{2}{7} for x_{2}.

$$7x^{2}-37x+10=7\left(x-5\right)\left(x-\frac{2}{7}\right)$$

Subtract \frac{2}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7x^{2}-37x+10=7\left(x-5\right)\times \frac{7x-2}{7}$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7x^{2}-37x+10=\left(x-5\right)\left(7x-2\right)$$

Evaluate

$$\left(x-5\right)\left(7x-2\right)$$