Solve 7x^2-60x+32

$$7 x ^ { 2 } - 60 x + 32$$

Factor

$$\left(x-8\right)\left(7x-4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx+32. To find a and b, set up a system to be solved.

$$a+b=-60 ab=7\times 32=224$$

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 224.

$$-1,-224 -2,-112 -4,-56 -7,-32 -8,-28 -14,-16$$

Calculate the sum for each pair.

$$-1-224=-225 -2-112=-114 -4-56=-60 -7-32=-39 -8-28=-36 -14-16=-30$$

The solution is the pair that gives sum -60.

$$a=-56 b=-4$$

Rewrite 7x^{2}-60x+32 as \left(7x^{2}-56x\right)+\left(-4x+32\right).

$$\left(7x^{2}-56x\right)+\left(-4x+32\right)$$

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

$$7x\left(x-8\right)-4\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(7x-4\right)$$

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.

$$7x^{2}-60x+32=0$$

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.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 7\times 32}}{2\times 7}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 7\times 32}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-60\right)±\sqrt{3600-28\times 32}}{2\times 7}$$

Multiply -28 times 32.

$$x=\frac{-\left(-60\right)±\sqrt{3600-896}}{2\times 7}$$

Add 3600 to -896.

$$x=\frac{-\left(-60\right)±\sqrt{2704}}{2\times 7}$$

Take the square root of 2704.

$$x=\frac{-\left(-60\right)±52}{2\times 7}$$

The opposite of -60 is 60.

$$x=\frac{60±52}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{60±52}{14}$$

Now solve the equation x=\frac{60±52}{14} when ± is plus. Add 60 to 52.

$$x=\frac{112}{14}$$

Divide 112 by 14.

$$x=8$$

Now solve the equation x=\frac{60±52}{14} when ± is minus. Subtract 52 from 60.

$$x=\frac{8}{14}$$

Reduce the fraction \frac{8}{14} to lowest terms by extracting and canceling out 2.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and \frac{4}{7} for x_{2}.

$$7x^{2}-60x+32=7\left(x-8\right)\left(x-\frac{4}{7}\right)$$

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

$$7x^{2}-60x+32=7\left(x-8\right)\times \frac{7x-4}{7}$$

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

$$7x^{2}-60x+32=\left(x-8\right)\left(7x-4\right)$$

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 7

$$x ^ 2 -\frac{60}{7}x +\frac{32}{7} = 0$$

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

$$r + s = \frac{60}{7} rs = \frac{32}{7}$$

Two numbers r and s sum up to \frac{60}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{60}{7} = \frac{30}{7}. 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>

$$r = \frac{30}{7} - u s = \frac{30}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{32}{7}

$$(\frac{30}{7} - u) (\frac{30}{7} + u) = \frac{32}{7}$$

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

$$\frac{900}{49} - u^2 = \frac{32}{7}$$

Simplify the expression by subtracting \frac{900}{49} on both sides

$$-u^2 = \frac{32}{7}-\frac{900}{49} = -\frac{676}{49}$$

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

$$u^2 = \frac{676}{49} u = \pm\sqrt{\frac{676}{49}} = \pm \frac{26}{7}$$

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

$$r =\frac{30}{7} - \frac{26}{7} = 0.571 s = \frac{30}{7} + \frac{26}{7} = 8$$

Evaluate

$$\left(x-8\right)\left(7x-4\right)$$