Solve 8x^2-6x-14

$$8 x ^ { 2 } - 6 x - 14$$

Factor

$$2\left(4x-7\right)\left(x+1\right)$$

Solution Steps

Factor out 2.

$$2\left(4x^{2}-3x-7\right)$$

Consider 4x^{2}-3x-7. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=-3 ab=4\left(-7\right)=-28$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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 -3.

$$a=-7 b=4$$

Rewrite 4x^{2}-3x-7 as \left(4x^{2}-7x\right)+\left(4x-7\right).

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

Factor out x in 4x^{2}-7x.

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

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

$$\left(4x-7\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$2\left(4x-7\right)\left(x+1\right)$$

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.

$$8x^{2}-6x-14=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 8\left(-14\right)}}{2\times 8}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 8\left(-14\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-6\right)±\sqrt{36-32\left(-14\right)}}{2\times 8}$$

Multiply -32 times -14.

$$x=\frac{-\left(-6\right)±\sqrt{36+448}}{2\times 8}$$

Add 36 to 448.

$$x=\frac{-\left(-6\right)±\sqrt{484}}{2\times 8}$$

Take the square root of 484.

$$x=\frac{-\left(-6\right)±22}{2\times 8}$$

The opposite of -6 is 6.

$$x=\frac{6±22}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{6±22}{16}$$

Now solve the equation x=\frac{6±22}{16} when ± is plus. Add 6 to 22.

$$x=\frac{28}{16}$$

Reduce the fraction \frac{28}{16} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{6±22}{16} when ± is minus. Subtract 22 from 6.

$$x=-\frac{16}{16}$$

Divide -16 by 16.

$$x=-1$$

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

$$8x^{2}-6x-14=8\left(x-\frac{7}{4}\right)\left(x-\left(-1\right)\right)$$

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

$$8x^{2}-6x-14=8\left(x-\frac{7}{4}\right)\left(x+1\right)$$

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

$$8x^{2}-6x-14=8\times \frac{4x-7}{4}\left(x+1\right)$$

Cancel out 4, the greatest common factor in 8 and 4.

$$8x^{2}-6x-14=2\left(4x-7\right)\left(x+1\right)$$

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 8

$$x ^ 2 -\frac{3}{4}x -\frac{7}{4} = 0$$

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

$$r + s = \frac{3}{4} rs = -\frac{7}{4}$$

Two numbers r and s sum up to \frac{3}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{3}{4} = \frac{3}{8}. 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>

$$r = \frac{3}{8} - u s = \frac{3}{8} + u$$

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

$$(\frac{3}{8} - u) (\frac{3}{8} + u) = -\frac{7}{4}$$

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

$$\frac{9}{64} - u^2 = -\frac{7}{4}$$

Simplify the expression by subtracting \frac{9}{64} on both sides

$$-u^2 = -\frac{7}{4}-\frac{9}{64} = -\frac{121}{64}$$

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

$$u^2 = \frac{121}{64} u = \pm\sqrt{\frac{121}{64}} = \pm \frac{11}{8}$$

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

$$r =\frac{3}{8} - \frac{11}{8} = -1 s = \frac{3}{8} + \frac{11}{8} = 1.750$$

Evaluate

$$2\left(4x-7\right)\left(x+1\right)$$