Solve -8x^2-15x+2

$$- 8 x ^ { 2 } - 15 x + 2$$

Factor

$$-\left(8x-1\right)\left(x+2\right)$$

Solution Steps

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

$$a+b=-15 ab=-8\times 2=-16$$

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 -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -15.

$$a=1 b=-16$$

Rewrite -8x^{2}-15x+2 as \left(-8x^{2}+x\right)+\left(-16x+2\right).

$$\left(-8x^{2}+x\right)+\left(-16x+2\right)$$

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

$$-x\left(8x-1\right)-2\left(8x-1\right)$$

Factor out common term 8x-1 by using distributive property.

$$\left(8x-1\right)\left(-x-2\right)$$

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.

$$-8x^{2}-15x+2=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-8\right)\times 2}}{2\left(-8\right)}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-8\right)\times 2}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-15\right)±\sqrt{225+32\times 2}}{2\left(-8\right)}$$

Multiply 32 times 2.

$$x=\frac{-\left(-15\right)±\sqrt{225+64}}{2\left(-8\right)}$$

Add 225 to 64.

$$x=\frac{-\left(-15\right)±\sqrt{289}}{2\left(-8\right)}$$

Take the square root of 289.

$$x=\frac{-\left(-15\right)±17}{2\left(-8\right)}$$

The opposite of -15 is 15.

$$x=\frac{15±17}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{15±17}{-16}$$

Now solve the equation x=\frac{15±17}{-16} when ± is plus. Add 15 to 17.

$$x=\frac{32}{-16}$$

Divide 32 by -16.

$$x=-2$$

Now solve the equation x=\frac{15±17}{-16} when ± is minus. Subtract 17 from 15.

$$x=-\frac{2}{-16}$$

Reduce the fraction \frac{-2}{-16} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and \frac{1}{8} for x_{2}.

$$-8x^{2}-15x+2=-8\left(x-\left(-2\right)\right)\left(x-\frac{1}{8}\right)$$

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

$$-8x^{2}-15x+2=-8\left(x+2\right)\left(x-\frac{1}{8}\right)$$

Subtract \frac{1}{8} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}-15x+2=-8\left(x+2\right)\times \frac{-8x+1}{-8}$$

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

$$-8x^{2}-15x+2=\left(x+2\right)\left(-8x+1\right)$$

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.

$$x ^ 2 +\frac{15}{8}x -\frac{1}{4} = 0$$

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

$$r + s = -\frac{15}{8} rs = -\frac{1}{4}$$

Two numbers r and s sum up to -\frac{15}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{15}{8} = -\frac{15}{16}. 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>

$$r = -\frac{15}{16} - u s = -\frac{15}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{4}

$$(-\frac{15}{16} - u) (-\frac{15}{16} + u) = -\frac{1}{4}$$

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

$$\frac{225}{256} - u^2 = -\frac{1}{4}$$

Simplify the expression by subtracting \frac{225}{256} on both sides

$$-u^2 = -\frac{1}{4}-\frac{225}{256} = -\frac{289}{256}$$

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

$$u^2 = \frac{289}{256} u = \pm\sqrt{\frac{289}{256}} = \pm \frac{17}{16}$$

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

$$r =-\frac{15}{16} - \frac{17}{16} = -2 s = -\frac{15}{16} + \frac{17}{16} = 0.125$$

Evaluate

$$-\left(8x-1\right)\left(x+2\right)$$