Solve -7x^2-17x+22

$$- 7 x ^ { 2 } - 17 x + 22$$

Factor

$$-7\left(x-\frac{-\sqrt{905}-17}{14}\right)\left(x-\frac{\sqrt{905}-17}{14}\right)$$

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.

$$-7x^{2}-17x+22=0$$

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.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\left(-7\right)\times 22}}{2\left(-7\right)}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\left(-7\right)\times 22}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-17\right)±\sqrt{289+28\times 22}}{2\left(-7\right)}$$

Multiply 28 times 22.

$$x=\frac{-\left(-17\right)±\sqrt{289+616}}{2\left(-7\right)}$$

Add 289 to 616.

$$x=\frac{-\left(-17\right)±\sqrt{905}}{2\left(-7\right)}$$

The opposite of -17 is 17.

$$x=\frac{17±\sqrt{905}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{17±\sqrt{905}}{-14}$$

Now solve the equation x=\frac{17±\sqrt{905}}{-14} when ± is plus. Add 17 to \sqrt{905}.

$$x=\frac{\sqrt{905}+17}{-14}$$

Divide 17+\sqrt{905} by -14.

$$x=\frac{-\sqrt{905}-17}{14}$$

Now solve the equation x=\frac{17±\sqrt{905}}{-14} when ± is minus. Subtract \sqrt{905} from 17.

$$x=\frac{17-\sqrt{905}}{-14}$$

Divide 17-\sqrt{905} by -14.

$$x=\frac{\sqrt{905}-17}{14}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-17-\sqrt{905}}{14} for x_{1} and \frac{-17+\sqrt{905}}{14} for x_{2}.

$$-7x^{2}-17x+22=-7\left(x-\frac{-\sqrt{905}-17}{14}\right)\left(x-\frac{\sqrt{905}-17}{14}\right)$$

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.

$$x ^ 2 +\frac{17}{7}x -\frac{22}{7} = 0$$

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

$$r + s = -\frac{17}{7} rs = -\frac{22}{7}$$

Two numbers r and s sum up to -\frac{17}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{7} = -\frac{17}{14}. 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>

$$r = -\frac{17}{14} - u s = -\frac{17}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{22}{7}

$$(-\frac{17}{14} - u) (-\frac{17}{14} + u) = -\frac{22}{7}$$

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

$$\frac{289}{196} - u^2 = -\frac{22}{7}$$

Simplify the expression by subtracting \frac{289}{196} on both sides

$$-u^2 = -\frac{22}{7}-\frac{289}{196} = -\frac{905}{196}$$

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

$$u^2 = \frac{905}{196} u = \pm\sqrt{\frac{905}{196}} = \pm \frac{\sqrt{905}}{14}$$

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

$$r =-\frac{17}{14} - \frac{\sqrt{905}}{14} = -3.363 s = -\frac{17}{14} + \frac{\sqrt{905}}{14} = 0.935$$

Evaluate

$$22-17x-7x^{2}$$