Solve 10x^2-10x-9

$$10 x ^ { 2 } - 10 x - 9$$

Factor

$$10\left(x-\left(-\frac{\sqrt{115}}{10}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{115}}{10}+\frac{1}{2}\right)\right)$$

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.

$$10x^{2}-10x-9=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 10\left(-9\right)}}{2\times 10}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 10\left(-9\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-10\right)±\sqrt{100-40\left(-9\right)}}{2\times 10}$$

Multiply -40 times -9.

$$x=\frac{-\left(-10\right)±\sqrt{100+360}}{2\times 10}$$

Add 100 to 360.

$$x=\frac{-\left(-10\right)±\sqrt{460}}{2\times 10}$$

Take the square root of 460.

$$x=\frac{-\left(-10\right)±2\sqrt{115}}{2\times 10}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{115}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{10±2\sqrt{115}}{20}$$

Now solve the equation x=\frac{10±2\sqrt{115}}{20} when ± is plus. Add 10 to 2\sqrt{115}.

$$x=\frac{2\sqrt{115}+10}{20}$$

Divide 10+2\sqrt{115} by 20.

$$x=\frac{\sqrt{115}}{10}+\frac{1}{2}$$

Now solve the equation x=\frac{10±2\sqrt{115}}{20} when ± is minus. Subtract 2\sqrt{115} from 10.

$$x=\frac{10-2\sqrt{115}}{20}$$

Divide 10-2\sqrt{115} by 20.

$$x=-\frac{\sqrt{115}}{10}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}+\frac{\sqrt{115}}{10} for x_{1} and \frac{1}{2}-\frac{\sqrt{115}}{10} for x_{2}.

$$10x^{2}-10x-9=10\left(x-\left(\frac{\sqrt{115}}{10}+\frac{1}{2}\right)\right)\left(x-\left(-\frac{\sqrt{115}}{10}+\frac{1}{2}\right)\right)$$

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.This is achieved by dividing both sides of the equation by 10

$$x ^ 2 -1x -\frac{9}{10} = 0$$

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

$$r + s = 1 rs = -\frac{9}{10}$$

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

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{10}

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -\frac{9}{10}$$

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

$$\frac{1}{4} - u^2 = -\frac{9}{10}$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -\frac{9}{10}-\frac{1}{4} = -\frac{23}{20}$$

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

$$u^2 = \frac{23}{20} u = \pm\sqrt{\frac{23}{20}} = \pm \frac{\sqrt{23}}{\sqrt{20}}$$

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

$$r =\frac{1}{2} - \frac{\sqrt{23}}{\sqrt{20}} = -0.572 s = \frac{1}{2} + \frac{\sqrt{23}}{\sqrt{20}} = 1.572$$

Evaluate

$$10x^{2}-10x-9$$