Solve 35m^2-69m+28

$$35 m ^ { 2 } - 69 m + 28$$

Factor

$$\left(5m-7\right)\left(7m-4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 35m^{2}+am+bm+28. To find a and b, set up a system to be solved.

$$a+b=-69 ab=35\times 28=980$$

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 980.

$$-1,-980 -2,-490 -4,-245 -5,-196 -7,-140 -10,-98 -14,-70 -20,-49 -28,-35$$

Calculate the sum for each pair.

$$-1-980=-981 -2-490=-492 -4-245=-249 -5-196=-201 -7-140=-147 -10-98=-108 -14-70=-84 -20-49=-69 -28-35=-63$$

The solution is the pair that gives sum -69.

$$a=-49 b=-20$$

Rewrite 35m^{2}-69m+28 as \left(35m^{2}-49m\right)+\left(-20m+28\right).

$$\left(35m^{2}-49m\right)+\left(-20m+28\right)$$

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

$$7m\left(5m-7\right)-4\left(5m-7\right)$$

Factor out common term 5m-7 by using distributive property.

$$\left(5m-7\right)\left(7m-4\right)$$

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.

$$35m^{2}-69m+28=0$$

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.

$$m=\frac{-\left(-69\right)±\sqrt{\left(-69\right)^{2}-4\times 35\times 28}}{2\times 35}$$

Square -69.

$$m=\frac{-\left(-69\right)±\sqrt{4761-4\times 35\times 28}}{2\times 35}$$

Multiply -4 times 35.

$$m=\frac{-\left(-69\right)±\sqrt{4761-140\times 28}}{2\times 35}$$

Multiply -140 times 28.

$$m=\frac{-\left(-69\right)±\sqrt{4761-3920}}{2\times 35}$$

Add 4761 to -3920.

$$m=\frac{-\left(-69\right)±\sqrt{841}}{2\times 35}$$

Take the square root of 841.

$$m=\frac{-\left(-69\right)±29}{2\times 35}$$

The opposite of -69 is 69.

$$m=\frac{69±29}{2\times 35}$$

Multiply 2 times 35.

$$m=\frac{69±29}{70}$$

Now solve the equation m=\frac{69±29}{70} when ± is plus. Add 69 to 29.

$$m=\frac{98}{70}$$

Reduce the fraction \frac{98}{70} to lowest terms by extracting and canceling out 14.

$$m=\frac{7}{5}$$

Now solve the equation m=\frac{69±29}{70} when ± is minus. Subtract 29 from 69.

$$m=\frac{40}{70}$$

Reduce the fraction \frac{40}{70} to lowest terms by extracting and canceling out 10.

$$m=\frac{4}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{5} for x_{1} and \frac{4}{7} for x_{2}.

$$35m^{2}-69m+28=35\left(m-\frac{7}{5}\right)\left(m-\frac{4}{7}\right)$$

Subtract \frac{7}{5} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$35m^{2}-69m+28=35\times \frac{5m-7}{5}\left(m-\frac{4}{7}\right)$$

Subtract \frac{4}{7} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$35m^{2}-69m+28=35\times \frac{5m-7}{5}\times \frac{7m-4}{7}$$

Multiply \frac{5m-7}{5} times \frac{7m-4}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$35m^{2}-69m+28=35\times \frac{\left(5m-7\right)\left(7m-4\right)}{5\times 7}$$

Multiply 5 times 7.

$$35m^{2}-69m+28=35\times \frac{\left(5m-7\right)\left(7m-4\right)}{35}$$

Cancel out 35, the greatest common factor in 35 and 35.

$$35m^{2}-69m+28=\left(5m-7\right)\left(7m-4\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 35

$$x ^ 2 -\frac{69}{35}x +\frac{4}{5} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{69}{35} rs = \frac{4}{5}$$

Two numbers r and s sum up to \frac{69}{35} exactly when the average of the two numbers is \frac{1}{2}*\frac{69}{35} = \frac{69}{70}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{69}{70} - u s = \frac{69}{70} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{5}

$$(\frac{69}{70} - u) (\frac{69}{70} + u) = \frac{4}{5}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{4761}{4900} - u^2 = \frac{4}{5}$$

Simplify the expression by subtracting \frac{4761}{4900} on both sides

$$-u^2 = \frac{4}{5}-\frac{4761}{4900} = -\frac{841}{4900}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{841}{4900} u = \pm\sqrt{\frac{841}{4900}} = \pm \frac{29}{70}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{69}{70} - \frac{29}{70} = 0.571 s = \frac{69}{70} + \frac{29}{70} = 1.400$$

Evaluate

$$\left(5m-7\right)\left(7m-4\right)$$