Solve 21x^2+37x+12

$$21 x ^ { 2 } + 37 x + 12$$

Factor

$$\left(3x+4\right)\left(7x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 21x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=37 ab=21\times 12=252$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 252.

$$1,252 2,126 3,84 4,63 6,42 7,36 9,28 12,21 14,18$$

Calculate the sum for each pair.

$$1+252=253 2+126=128 3+84=87 4+63=67 6+42=48 7+36=43 9+28=37 12+21=33 14+18=32$$

The solution is the pair that gives sum 37.

$$a=9 b=28$$

Rewrite 21x^{2}+37x+12 as \left(21x^{2}+9x\right)+\left(28x+12\right).

$$\left(21x^{2}+9x\right)+\left(28x+12\right)$$

Factor out 3x in the first and 4 in the second group.

$$3x\left(7x+3\right)+4\left(7x+3\right)$$

Factor out common term 7x+3 by using distributive property.

$$\left(7x+3\right)\left(3x+4\right)$$

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.

$$21x^{2}+37x+12=0$$

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.

$$x=\frac{-37±\sqrt{37^{2}-4\times 21\times 12}}{2\times 21}$$

Square 37.

$$x=\frac{-37±\sqrt{1369-4\times 21\times 12}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-37±\sqrt{1369-84\times 12}}{2\times 21}$$

Multiply -84 times 12.

$$x=\frac{-37±\sqrt{1369-1008}}{2\times 21}$$

Add 1369 to -1008.

$$x=\frac{-37±\sqrt{361}}{2\times 21}$$

Take the square root of 361.

$$x=\frac{-37±19}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{-37±19}{42}$$

Now solve the equation x=\frac{-37±19}{42} when ± is plus. Add -37 to 19.

$$x=-\frac{18}{42}$$

Reduce the fraction \frac{-18}{42} to lowest terms by extracting and canceling out 6.

$$x=-\frac{3}{7}$$

Now solve the equation x=\frac{-37±19}{42} when ± is minus. Subtract 19 from -37.

$$x=-\frac{56}{42}$$

Reduce the fraction \frac{-56}{42} to lowest terms by extracting and canceling out 14.

$$x=-\frac{4}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{7} for x_{1} and -\frac{4}{3} for x_{2}.

$$21x^{2}+37x+12=21\left(x-\left(-\frac{3}{7}\right)\right)\left(x-\left(-\frac{4}{3}\right)\right)$$

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

$$21x^{2}+37x+12=21\left(x+\frac{3}{7}\right)\left(x+\frac{4}{3}\right)$$

Add \frac{3}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$21x^{2}+37x+12=21\times \frac{7x+3}{7}\left(x+\frac{4}{3}\right)$$

Add \frac{4}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$21x^{2}+37x+12=21\times \frac{7x+3}{7}\times \frac{3x+4}{3}$$

Multiply \frac{7x+3}{7} times \frac{3x+4}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$21x^{2}+37x+12=21\times \frac{\left(7x+3\right)\left(3x+4\right)}{7\times 3}$$

Multiply 7 times 3.

$$21x^{2}+37x+12=21\times \frac{\left(7x+3\right)\left(3x+4\right)}{21}$$

Cancel out 21, the greatest common factor in 21 and 21.

$$21x^{2}+37x+12=\left(7x+3\right)\left(3x+4\right)$$

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 21

$$x ^ 2 +\frac{37}{21}x +\frac{4}{7} = 0$$

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

$$r + s = -\frac{37}{21} rs = \frac{4}{7}$$

Two numbers r and s sum up to -\frac{37}{21} exactly when the average of the two numbers is \frac{1}{2}*-\frac{37}{21} = -\frac{37}{42}. 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>

$$r = -\frac{37}{42} - u s = -\frac{37}{42} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{7}

$$(-\frac{37}{42} - u) (-\frac{37}{42} + u) = \frac{4}{7}$$

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

$$\frac{1369}{1764} - u^2 = \frac{4}{7}$$

Simplify the expression by subtracting \frac{1369}{1764} on both sides

$$-u^2 = \frac{4}{7}-\frac{1369}{1764} = -\frac{361}{1764}$$

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

$$u^2 = \frac{361}{1764} u = \pm\sqrt{\frac{361}{1764}} = \pm \frac{19}{42}$$

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

$$r =-\frac{37}{42} - \frac{19}{42} = -1.333 s = -\frac{37}{42} + \frac{19}{42} = -0.429$$

Evaluate

$$\left(3x+4\right)\left(7x+3\right)$$