Solve 3x^2-36x+105

$$3 x ^ { 2 } - 36 x + 105$$

Factor

$$3\left(x-7\right)\left(x-5\right)$$

Solution Steps

Factor out 3.

$$3\left(x^{2}-12x+35\right)$$

Consider x^{2}-12x+35. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+35. To find a and b, set up a system to be solved.

$$a+b=-12 ab=1\times 35=35$$

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 35.

$$-1,-35 -5,-7$$

Calculate the sum for each pair.

$$-1-35=-36 -5-7=-12$$

The solution is the pair that gives sum -12.

$$a=-7 b=-5$$

Rewrite x^{2}-12x+35 as \left(x^{2}-7x\right)+\left(-5x+35\right).

$$\left(x^{2}-7x\right)+\left(-5x+35\right)$$

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

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

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

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

Rewrite the complete factored expression.

$$3\left(x-7\right)\left(x-5\right)$$

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.

$$3x^{2}-36x+105=0$$

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.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 3\times 105}}{2\times 3}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 3\times 105}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-36\right)±\sqrt{1296-12\times 105}}{2\times 3}$$

Multiply -12 times 105.

$$x=\frac{-\left(-36\right)±\sqrt{1296-1260}}{2\times 3}$$

Add 1296 to -1260.

$$x=\frac{-\left(-36\right)±\sqrt{36}}{2\times 3}$$

Take the square root of 36.

$$x=\frac{-\left(-36\right)±6}{2\times 3}$$

The opposite of -36 is 36.

$$x=\frac{36±6}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{36±6}{6}$$

Now solve the equation x=\frac{36±6}{6} when ± is plus. Add 36 to 6.

$$x=\frac{42}{6}$$

Divide 42 by 6.

$$x=7$$

Now solve the equation x=\frac{36±6}{6} when ± is minus. Subtract 6 from 36.

$$x=\frac{30}{6}$$

Divide 30 by 6.

$$x=5$$

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

$$3x^{2}-36x+105=3\left(x-7\right)\left(x-5\right)$$

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 3

$$x ^ 2 -12x +35 = 0$$

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

$$r + s = 12 rs = 35$$

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

$$r = 6 - u s = 6 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 35

$$(6 - u) (6 + u) = 35$$

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

$$36 - u^2 = 35$$

Simplify the expression by subtracting 36 on both sides

$$-u^2 = 35-36 = -1$$

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

$$u^2 = 1 u = \pm\sqrt{1} = \pm 1$$

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

$$r =6 - 1 = 5 s = 6 + 1 = 7$$

Evaluate

$$3\left(x-7\right)\left(x-5\right)$$