Solve 21x^2+54x+24

$$21 x ^ { 2 } + 54 x + 24$$

Factor

$$3\left(x+2\right)\left(7x+4\right)$$

Solution Steps

Factor out 3.

$$3\left(7x^{2}+18x+8\right)$$

Consider 7x^{2}+18x+8. Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=18 ab=7\times 8=56$$

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 56.

$$1,56 2,28 4,14 7,8$$

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 18.

$$a=4 b=14$$

Rewrite 7x^{2}+18x+8 as \left(7x^{2}+4x\right)+\left(14x+8\right).

$$\left(7x^{2}+4x\right)+\left(14x+8\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(7x+4\right)+2\left(7x+4\right)$$

Factor out common term 7x+4 by using distributive property.

$$\left(7x+4\right)\left(x+2\right)$$

Rewrite the complete factored expression.

$$3\left(7x+4\right)\left(x+2\right)$$

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.

$$21x^{2}+54x+24=0$$

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.

$$x=\frac{-54±\sqrt{54^{2}-4\times 21\times 24}}{2\times 21}$$

Square 54.

$$x=\frac{-54±\sqrt{2916-4\times 21\times 24}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-54±\sqrt{2916-84\times 24}}{2\times 21}$$

Multiply -84 times 24.

$$x=\frac{-54±\sqrt{2916-2016}}{2\times 21}$$

Add 2916 to -2016.

$$x=\frac{-54±\sqrt{900}}{2\times 21}$$

Take the square root of 900.

$$x=\frac{-54±30}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{-54±30}{42}$$

Now solve the equation x=\frac{-54±30}{42} when ± is plus. Add -54 to 30.

$$x=-\frac{24}{42}$$

Reduce the fraction \frac{-24}{42} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{-54±30}{42} when ± is minus. Subtract 30 from -54.

$$x=-\frac{84}{42}$$

Divide -84 by 42.

$$x=-2$$

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

$$21x^{2}+54x+24=21\left(x-\left(-\frac{4}{7}\right)\right)\left(x-\left(-2\right)\right)$$

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

$$21x^{2}+54x+24=21\left(x+\frac{4}{7}\right)\left(x+2\right)$$

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

$$21x^{2}+54x+24=21\times \frac{7x+4}{7}\left(x+2\right)$$

Cancel out 7, the greatest common factor in 21 and 7.

$$21x^{2}+54x+24=3\left(7x+4\right)\left(x+2\right)$$

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

$$x ^ 2 +\frac{18}{7}x +\frac{8}{7} = 0$$

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

$$r + s = -\frac{18}{7} rs = \frac{8}{7}$$

Two numbers r and s sum up to -\frac{18}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{18}{7} = -\frac{9}{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>

$$r = -\frac{9}{7} - u s = -\frac{9}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{7}

$$(-\frac{9}{7} - u) (-\frac{9}{7} + u) = \frac{8}{7}$$

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

$$\frac{81}{49} - u^2 = \frac{8}{7}$$

Simplify the expression by subtracting \frac{81}{49} on both sides

$$-u^2 = \frac{8}{7}-\frac{81}{49} = -\frac{25}{49}$$

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

$$u^2 = \frac{25}{49} u = \pm\sqrt{\frac{25}{49}} = \pm \frac{5}{7}$$

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

$$r =-\frac{9}{7} - \frac{5}{7} = -2.000 s = -\frac{9}{7} + \frac{5}{7} = -0.571$$

Evaluate

$$3\left(x+2\right)\left(7x+4\right)$$