Solve -2x^2-17x+30

$$- 2 x ^ { 2 } - 17 x + 30$$

Factor

$$-\left(2x-3\right)\left(x+10\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+30. To find a and b, set up a system to be solved.

$$a+b=-17 ab=-2\times 30=-60$$

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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -17.

$$a=3 b=-20$$

Rewrite -2x^{2}-17x+30 as \left(-2x^{2}+3x\right)+\left(-20x+30\right).

$$\left(-2x^{2}+3x\right)+\left(-20x+30\right)$$

Factor out -x in the first and -10 in the second group.

$$-x\left(2x-3\right)-10\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(-x-10\right)$$

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.

$$-2x^{2}-17x+30=0$$

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.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\left(-2\right)\times 30}}{2\left(-2\right)}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\left(-2\right)\times 30}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-17\right)±\sqrt{289+8\times 30}}{2\left(-2\right)}$$

Multiply 8 times 30.

$$x=\frac{-\left(-17\right)±\sqrt{289+240}}{2\left(-2\right)}$$

Add 289 to 240.

$$x=\frac{-\left(-17\right)±\sqrt{529}}{2\left(-2\right)}$$

Take the square root of 529.

$$x=\frac{-\left(-17\right)±23}{2\left(-2\right)}$$

The opposite of -17 is 17.

$$x=\frac{17±23}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{17±23}{-4}$$

Now solve the equation x=\frac{17±23}{-4} when ± is plus. Add 17 to 23.

$$x=\frac{40}{-4}$$

Divide 40 by -4.

$$x=-10$$

Now solve the equation x=\frac{17±23}{-4} when ± is minus. Subtract 23 from 17.

$$x=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -10 for x_{1} and \frac{3}{2} for x_{2}.

$$-2x^{2}-17x+30=-2\left(x-\left(-10\right)\right)\left(x-\frac{3}{2}\right)$$

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

$$-2x^{2}-17x+30=-2\left(x+10\right)\left(x-\frac{3}{2}\right)$$

Subtract \frac{3}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-2x^{2}-17x+30=-2\left(x+10\right)\times \frac{-2x+3}{-2}$$

Cancel out 2, the greatest common factor in -2 and 2.

$$-2x^{2}-17x+30=\left(x+10\right)\left(-2x+3\right)$$

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.

$$x ^ 2 +\frac{17}{2}x -15 = 0$$

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

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

Two numbers r and s sum up to -\frac{17}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{2} = -\frac{17}{4}. 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>

$$r = -\frac{17}{4} - u s = -\frac{17}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -15

$$(-\frac{17}{4} - u) (-\frac{17}{4} + u) = -15$$

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

$$\frac{289}{16} - u^2 = -15$$

Simplify the expression by subtracting \frac{289}{16} on both sides

$$-u^2 = -15-\frac{289}{16} = -\frac{529}{16}$$

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

$$u^2 = \frac{529}{16} u = \pm\sqrt{\frac{529}{16}} = \pm \frac{23}{4}$$

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

$$r =-\frac{17}{4} - \frac{23}{4} = -10 s = -\frac{17}{4} + \frac{23}{4} = 1.500$$

Evaluate

$$-\left(2x-3\right)\left(x+10\right)$$