Solve 35x^2-12x-32

$$35 x ^ { 2 } - 12 x - 32$$

Factor

$$\left(7x-8\right)\left(5x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 35x^{2}+ax+bx-32. To find a and b, set up a system to be solved.

$$a+b=-12 ab=35\left(-32\right)=-1120$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1120.

$$1,-1120 2,-560 4,-280 5,-224 7,-160 8,-140 10,-112 14,-80 16,-70 20,-56 28,-40 32,-35$$

Calculate the sum for each pair.

$$1-1120=-1119 2-560=-558 4-280=-276 5-224=-219 7-160=-153 8-140=-132 10-112=-102 14-80=-66 16-70=-54 20-56=-36 28-40=-12 32-35=-3$$

The solution is the pair that gives sum -12.

$$a=-40 b=28$$

Rewrite 35x^{2}-12x-32 as \left(35x^{2}-40x\right)+\left(28x-32\right).

$$\left(35x^{2}-40x\right)+\left(28x-32\right)$$

Factor out 5x in the first and 4 in the second group.

$$5x\left(7x-8\right)+4\left(7x-8\right)$$

Factor out common term 7x-8 by using distributive property.

$$\left(7x-8\right)\left(5x+4\right)$$

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.

$$35x^{2}-12x-32=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 35\left(-32\right)}}{2\times 35}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 35\left(-32\right)}}{2\times 35}$$

Multiply -4 times 35.

$$x=\frac{-\left(-12\right)±\sqrt{144-140\left(-32\right)}}{2\times 35}$$

Multiply -140 times -32.

$$x=\frac{-\left(-12\right)±\sqrt{144+4480}}{2\times 35}$$

Add 144 to 4480.

$$x=\frac{-\left(-12\right)±\sqrt{4624}}{2\times 35}$$

Take the square root of 4624.

$$x=\frac{-\left(-12\right)±68}{2\times 35}$$

The opposite of -12 is 12.

$$x=\frac{12±68}{2\times 35}$$

Multiply 2 times 35.

$$x=\frac{12±68}{70}$$

Now solve the equation x=\frac{12±68}{70} when ± is plus. Add 12 to 68.

$$x=\frac{80}{70}$$

Reduce the fraction \frac{80}{70} to lowest terms by extracting and canceling out 10.

$$x=\frac{8}{7}$$

Now solve the equation x=\frac{12±68}{70} when ± is minus. Subtract 68 from 12.

$$x=-\frac{56}{70}$$

Reduce the fraction \frac{-56}{70} to lowest terms by extracting and canceling out 14.

$$x=-\frac{4}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{8}{7} for x_{1} and -\frac{4}{5} for x_{2}.

$$35x^{2}-12x-32=35\left(x-\frac{8}{7}\right)\left(x-\left(-\frac{4}{5}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$35x^{2}-12x-32=35\left(x-\frac{8}{7}\right)\left(x+\frac{4}{5}\right)$$

Subtract \frac{8}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$35x^{2}-12x-32=35\times \frac{7x-8}{7}\left(x+\frac{4}{5}\right)$$

Add \frac{4}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$35x^{2}-12x-32=35\times \frac{7x-8}{7}\times \frac{5x+4}{5}$$

Multiply \frac{7x-8}{7} times \frac{5x+4}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$35x^{2}-12x-32=35\times \frac{\left(7x-8\right)\left(5x+4\right)}{7\times 5}$$

Multiply 7 times 5.

$$35x^{2}-12x-32=35\times \frac{\left(7x-8\right)\left(5x+4\right)}{35}$$

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

$$35x^{2}-12x-32=\left(7x-8\right)\left(5x+4\right)$$

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

$$x ^ 2 -\frac{12}{35}x -\frac{32}{35} = 0$$

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

$$r + s = \frac{12}{35} rs = -\frac{32}{35}$$

Two numbers r and s sum up to \frac{12}{35} exactly when the average of the two numbers is \frac{1}{2}*\frac{12}{35} = \frac{6}{35}. 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>

$$r = \frac{6}{35} - u s = \frac{6}{35} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{32}{35}

$$(\frac{6}{35} - u) (\frac{6}{35} + u) = -\frac{32}{35}$$

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

$$\frac{36}{1225} - u^2 = -\frac{32}{35}$$

Simplify the expression by subtracting \frac{36}{1225} on both sides

$$-u^2 = -\frac{32}{35}-\frac{36}{1225} = -\frac{1156}{1225}$$

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

$$u^2 = \frac{1156}{1225} u = \pm\sqrt{\frac{1156}{1225}} = \pm \frac{34}{35}$$

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

$$r =\frac{6}{35} - \frac{34}{35} = -0.800 s = \frac{6}{35} + \frac{34}{35} = 1.143$$

Evaluate

$$\left(7x-8\right)\left(5x+4\right)$$