Solve 7y^2+12y-4

$$7 y ^ { 2 } + 12 y - 4$$

Factor

$$\left(7y-2\right)\left(y+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7y^{2}+ay+by-4. To find a and b, set up a system to be solved.

$$a+b=12 ab=7\left(-4\right)=-28$$

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 -28.

$$-1,28 -2,14 -4,7$$

Calculate the sum for each pair.

$$-1+28=27 -2+14=12 -4+7=3$$

The solution is the pair that gives sum 12.

$$a=-2 b=14$$

Rewrite 7y^{2}+12y-4 as \left(7y^{2}-2y\right)+\left(14y-4\right).

$$\left(7y^{2}-2y\right)+\left(14y-4\right)$$

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

$$y\left(7y-2\right)+2\left(7y-2\right)$$

Factor out common term 7y-2 by using distributive property.

$$\left(7y-2\right)\left(y+2\right)$$

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.

$$7y^{2}+12y-4=0$$

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.

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

Square 12.

$$y=\frac{-12±\sqrt{144-4\times 7\left(-4\right)}}{2\times 7}$$

Multiply -4 times 7.

$$y=\frac{-12±\sqrt{144-28\left(-4\right)}}{2\times 7}$$

Multiply -28 times -4.

$$y=\frac{-12±\sqrt{144+112}}{2\times 7}$$

Add 144 to 112.

$$y=\frac{-12±\sqrt{256}}{2\times 7}$$

Take the square root of 256.

$$y=\frac{-12±16}{2\times 7}$$

Multiply 2 times 7.

$$y=\frac{-12±16}{14}$$

Now solve the equation y=\frac{-12±16}{14} when ± is plus. Add -12 to 16.

$$y=\frac{4}{14}$$

Reduce the fraction \frac{4}{14} to lowest terms by extracting and canceling out 2.

$$y=\frac{2}{7}$$

Now solve the equation y=\frac{-12±16}{14} when ± is minus. Subtract 16 from -12.

$$y=-\frac{28}{14}$$

Divide -28 by 14.

$$y=-2$$

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

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

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

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

Subtract \frac{2}{7} from y by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7y^{2}+12y-4=7\times \frac{7y-2}{7}\left(y+2\right)$$

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

$$7y^{2}+12y-4=\left(7y-2\right)\left(y+2\right)$$

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 7

$$x ^ 2 +\frac{12}{7}x -\frac{4}{7} = 0$$

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

$$r + s = -\frac{12}{7} rs = -\frac{4}{7}$$

Two numbers r and s sum up to -\frac{12}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{12}{7} = -\frac{6}{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>

$$r = -\frac{6}{7} - u s = -\frac{6}{7} + u$$

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

$$(-\frac{6}{7} - u) (-\frac{6}{7} + u) = -\frac{4}{7}$$

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

$$\frac{36}{49} - u^2 = -\frac{4}{7}$$

Simplify the expression by subtracting \frac{36}{49} on both sides

$$-u^2 = -\frac{4}{7}-\frac{36}{49} = -\frac{64}{49}$$

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

$$u^2 = \frac{64}{49} u = \pm\sqrt{\frac{64}{49}} = \pm \frac{8}{7}$$

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

$$r =-\frac{6}{7} - \frac{8}{7} = -2 s = -\frac{6}{7} + \frac{8}{7} = 0.286$$

Evaluate

$$\left(7y-2\right)\left(y+2\right)$$