Solve -8x^2-9x+17

$$- 8 x ^ { 2 } - 9 x + 17$$

Factor

$$\left(1-x\right)\left(8x+17\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -8x^{2}+ax+bx+17. To find a and b, set up a system to be solved.

$$a+b=-9 ab=-8\times 17=-136$$

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 -136.

$$1,-136 2,-68 4,-34 8,-17$$

Calculate the sum for each pair.

$$1-136=-135 2-68=-66 4-34=-30 8-17=-9$$

The solution is the pair that gives sum -9.

$$a=8 b=-17$$

Rewrite -8x^{2}-9x+17 as \left(-8x^{2}+8x\right)+\left(-17x+17\right).

$$\left(-8x^{2}+8x\right)+\left(-17x+17\right)$$

Factor out 8x in the first and 17 in the second group.

$$8x\left(-x+1\right)+17\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(8x+17\right)$$

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.

$$-8x^{2}-9x+17=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-8\right)\times 17}}{2\left(-8\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-8\right)\times 17}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-9\right)±\sqrt{81+32\times 17}}{2\left(-8\right)}$$

Multiply 32 times 17.

$$x=\frac{-\left(-9\right)±\sqrt{81+544}}{2\left(-8\right)}$$

Add 81 to 544.

$$x=\frac{-\left(-9\right)±\sqrt{625}}{2\left(-8\right)}$$

Take the square root of 625.

$$x=\frac{-\left(-9\right)±25}{2\left(-8\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±25}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{9±25}{-16}$$

Now solve the equation x=\frac{9±25}{-16} when ± is plus. Add 9 to 25.

$$x=\frac{34}{-16}$$

Reduce the fraction \frac{34}{-16} to lowest terms by extracting and canceling out 2.

$$x=-\frac{17}{8}$$

Now solve the equation x=\frac{9±25}{-16} when ± is minus. Subtract 25 from 9.

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

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{17}{8} for x_{1} and 1 for x_{2}.

$$-8x^{2}-9x+17=-8\left(x-\left(-\frac{17}{8}\right)\right)\left(x-1\right)$$

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

$$-8x^{2}-9x+17=-8\left(x+\frac{17}{8}\right)\left(x-1\right)$$

Add \frac{17}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}-9x+17=-8\times \frac{-8x-17}{-8}\left(x-1\right)$$

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

$$-8x^{2}-9x+17=\left(-8x-17\right)\left(x-1\right)$$

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.

$$x ^ 2 +\frac{9}{8}x -\frac{17}{8} = 0$$

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

$$r + s = -\frac{9}{8} rs = -\frac{17}{8}$$

Two numbers r and s sum up to -\frac{9}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{9}{8} = -\frac{9}{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>

$$r = -\frac{9}{16} - u s = -\frac{9}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{17}{8}

$$(-\frac{9}{16} - u) (-\frac{9}{16} + u) = -\frac{17}{8}$$

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

$$\frac{81}{256} - u^2 = -\frac{17}{8}$$

Simplify the expression by subtracting \frac{81}{256} on both sides

$$-u^2 = -\frac{17}{8}-\frac{81}{256} = -\frac{625}{256}$$

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

$$u^2 = \frac{625}{256} u = \pm\sqrt{\frac{625}{256}} = \pm \frac{25}{16}$$

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

$$r =-\frac{9}{16} - \frac{25}{16} = -2.125 s = -\frac{9}{16} + \frac{25}{16} = 1$$

Evaluate

$$\left(1-x\right)\left(8x+17\right)$$