Solve 4r^2-7r-28r+49

$$4 r ^ { 2 } - 7 r - 28 r + 49$$

Factor

$$\left(r-7\right)\left(4r-7\right)$$

Solution Steps

Multiply and combine like terms.

$$4r^{2}-35r+49$$

Factor the expression by grouping. First, the expression needs to be rewritten as 4r^{2}+ar+br+49. To find a and b, set up a system to be solved.

$$a+b=-35 ab=4\times 49=196$$

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 196.

$$-1,-196 -2,-98 -4,-49 -7,-28 -14,-14$$

Calculate the sum for each pair.

$$-1-196=-197 -2-98=-100 -4-49=-53 -7-28=-35 -14-14=-28$$

The solution is the pair that gives sum -35.

$$a=-28 b=-7$$

Rewrite 4r^{2}-35r+49 as \left(4r^{2}-28r\right)+\left(-7r+49\right).

$$\left(4r^{2}-28r\right)+\left(-7r+49\right)$$

Factor out 4r in the first and -7 in the second group.

$$4r\left(r-7\right)-7\left(r-7\right)$$

Factor out common term r-7 by using distributive property.

$$\left(r-7\right)\left(4r-7\right)$$

Evaluate

$$\left(r-7\right)\left(4r-7\right)$$

Solution Steps

Combine -7r and -28r to get -35r.

$$4r^{2}-35r+49$$