Solve 7x^2+25x+12

$$7 x ^ { 2 } + 25 x + 12$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=25 ab=7\times 12=84$$

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 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Calculate the sum for each pair.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

The solution is the pair that gives sum 25.

$$a=4 b=21$$

Rewrite 7x^{2}+25x+12 as \left(7x^{2}+4x\right)+\left(21x+12\right).

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

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

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

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

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

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.

$$7x^{2}+25x+12=0$$

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.

$$x=\frac{-25±\sqrt{25^{2}-4\times 7\times 12}}{2\times 7}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 7\times 12}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-25±\sqrt{625-28\times 12}}{2\times 7}$$

Multiply -28 times 12.

$$x=\frac{-25±\sqrt{625-336}}{2\times 7}$$

Add 625 to -336.

$$x=\frac{-25±\sqrt{289}}{2\times 7}$$

Take the square root of 289.

$$x=\frac{-25±17}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-25±17}{14}$$

Now solve the equation x=\frac{-25±17}{14} when ± is plus. Add -25 to 17.

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

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

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

Now solve the equation x=\frac{-25±17}{14} when ± is minus. Subtract 17 from -25.

$$x=-\frac{42}{14}$$

Divide -42 by 14.

$$x=-3$$

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 -3 for x_{2}.

$$7x^{2}+25x+12=7\left(x-\left(-\frac{4}{7}\right)\right)\left(x-\left(-3\right)\right)$$

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

$$7x^{2}+25x+12=7\left(x+\frac{4}{7}\right)\left(x+3\right)$$

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

$$7x^{2}+25x+12=7\times \frac{7x+4}{7}\left(x+3\right)$$

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

$$7x^{2}+25x+12=\left(7x+4\right)\left(x+3\right)$$

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

$$x ^ 2 +\frac{25}{7}x +\frac{12}{7} = 0$$

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

$$r + s = -\frac{25}{7} rs = \frac{12}{7}$$

Two numbers r and s sum up to -\frac{25}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{25}{7} = -\frac{25}{14}. 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>

$$r = -\frac{25}{14} - u s = -\frac{25}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{12}{7}

$$(-\frac{25}{14} - u) (-\frac{25}{14} + u) = \frac{12}{7}$$

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

$$\frac{625}{196} - u^2 = \frac{12}{7}$$

Simplify the expression by subtracting \frac{625}{196} on both sides

$$-u^2 = \frac{12}{7}-\frac{625}{196} = -\frac{289}{196}$$

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

$$u^2 = \frac{289}{196} u = \pm\sqrt{\frac{289}{196}} = \pm \frac{17}{14}$$

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

$$r =-\frac{25}{14} - \frac{17}{14} = -3 s = -\frac{25}{14} + \frac{17}{14} = -0.571$$

Evaluate

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