Solve -x^2-6x+10

$$- x ^ { 2 } - 6 x + 10$$

Factor

$$-\left(x-\left(-\sqrt{19}-3\right)\right)\left(x-\left(\sqrt{19}-3\right)\right)$$

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.

$$-x^{2}-6x+10=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 10}}{2\left(-1\right)}$$

Multiply 4 times 10.

$$x=\frac{-\left(-6\right)±\sqrt{36+40}}{2\left(-1\right)}$$

Add 36 to 40.

$$x=\frac{-\left(-6\right)±\sqrt{76}}{2\left(-1\right)}$$

Take the square root of 76.

$$x=\frac{-\left(-6\right)±2\sqrt{19}}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{19}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±2\sqrt{19}}{-2}$$

Now solve the equation x=\frac{6±2\sqrt{19}}{-2} when ± is plus. Add 6 to 2\sqrt{19}.

$$x=\frac{2\sqrt{19}+6}{-2}$$

Divide 6+2\sqrt{19} by -2.

$$x=-\left(\sqrt{19}+3\right)$$

Now solve the equation x=\frac{6±2\sqrt{19}}{-2} when ± is minus. Subtract 2\sqrt{19} from 6.

$$x=\frac{6-2\sqrt{19}}{-2}$$

Divide 6-2\sqrt{19} by -2.

$$x=\sqrt{19}-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\left(3+\sqrt{19}\right) for x_{1} and -3+\sqrt{19} for x_{2}.

$$-x^{2}-6x+10=-\left(x-\left(-\left(\sqrt{19}+3\right)\right)\right)\left(x-\left(\sqrt{19}-3\right)\right)$$

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.

$$x ^ 2 +6x -10 = 0$$

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

$$r + s = -6 rs = -10$$

Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -3. 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>

$$r = -3 - u s = -3 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -10

$$(-3 - u) (-3 + u) = -10$$

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

$$9 - u^2 = -10$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = -10-9 = -19$$

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

$$u^2 = 19 u = \pm\sqrt{19} = \pm \sqrt{19}$$

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

$$r =-3 - \sqrt{19} = -7.359 s = -3 + \sqrt{19} = 1.359$$

Evaluate

$$10-6x-x^{2}$$