Solve 7x^2+5x-12

$$7 x ^ { 2 } + 5 x - 12$$

Factor

$$\left(x-1\right)\left(7x+12\right)$$

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.

$$a+b=5 ab=7\left(-12\right)=-84$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. 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.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 5.

$$a=-7 b=12$$

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

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

Factor out 7x in the first and 12 in the second group.

$$7x\left(x-1\right)+12\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(7x+12\right)$$

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.

$$7x^{2}+5x-12=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 7\left(-12\right)}}{2\times 7}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 7\left(-12\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-5±\sqrt{25-28\left(-12\right)}}{2\times 7}$$

Multiply -28 times -12.

$$x=\frac{-5±\sqrt{25+336}}{2\times 7}$$

Add 25 to 336.

$$x=\frac{-5±\sqrt{361}}{2\times 7}$$

Take the square root of 361.

$$x=\frac{-5±19}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-5±19}{14}$$

Now solve the equation x=\frac{-5±19}{14} when ± is plus. Add -5 to 19.

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

Divide 14 by 14.

$$x=1$$

Now solve the equation x=\frac{-5±19}{14} when ± is minus. Subtract 19 from -5.

$$x=-\frac{24}{14}$$

Reduce the fraction \frac{-24}{14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{12}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -\frac{12}{7} for x_{2}.

$$7x^{2}+5x-12=7\left(x-1\right)\left(x-\left(-\frac{12}{7}\right)\right)$$

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

$$7x^{2}+5x-12=7\left(x-1\right)\left(x+\frac{12}{7}\right)$$

Add \frac{12}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7x^{2}+5x-12=7\left(x-1\right)\times \frac{7x+12}{7}$$

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

$$7x^{2}+5x-12=\left(x-1\right)\left(7x+12\right)$$

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

$$x ^ 2 +\frac{5}{7}x -\frac{12}{7} = 0$$

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

$$r + s = -\frac{5}{7} rs = -\frac{12}{7}$$

Two numbers r and s sum up to -\frac{5}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{5}{7} = -\frac{5}{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>

$$r = -\frac{5}{14} - u s = -\frac{5}{14} + u$$

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

$$(-\frac{5}{14} - u) (-\frac{5}{14} + u) = -\frac{12}{7}$$

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

$$\frac{25}{196} - u^2 = -\frac{12}{7}$$

Simplify the expression by subtracting \frac{25}{196} on both sides

$$-u^2 = -\frac{12}{7}-\frac{25}{196} = -\frac{361}{196}$$

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

$$u^2 = \frac{361}{196} u = \pm\sqrt{\frac{361}{196}} = \pm \frac{19}{14}$$

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

$$r =-\frac{5}{14} - \frac{19}{14} = -1.714 s = -\frac{5}{14} + \frac{19}{14} = 1$$

Evaluate

$$\left(x-1\right)\left(7x+12\right)$$