Solve 10x^2-x-28

$$10 x ^ { 2 } - x - 28$$

Factor

$$10\left(x-\frac{1-\sqrt{1121}}{20}\right)\left(x-\frac{\sqrt{1121}+1}{20}\right)$$

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.

$$10x^{2}-x-28=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 10\left(-28\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-1\right)±\sqrt{1-40\left(-28\right)}}{2\times 10}$$

Multiply -40 times -28.

$$x=\frac{-\left(-1\right)±\sqrt{1+1120}}{2\times 10}$$

Add 1 to 1120.

$$x=\frac{-\left(-1\right)±\sqrt{1121}}{2\times 10}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{1121}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{1±\sqrt{1121}}{20}$$

Now solve the equation x=\frac{1±\sqrt{1121}}{20} when ± is plus. Add 1 to \sqrt{1121}.

$$x=\frac{\sqrt{1121}+1}{20}$$

Now solve the equation x=\frac{1±\sqrt{1121}}{20} when ± is minus. Subtract \sqrt{1121} from 1.

$$x=\frac{1-\sqrt{1121}}{20}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{1121}}{20} for x_{1} and \frac{1-\sqrt{1121}}{20} for x_{2}.

$$10x^{2}-x-28=10\left(x-\frac{\sqrt{1121}+1}{20}\right)\left(x-\frac{1-\sqrt{1121}}{20}\right)$$

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

$$x ^ 2 -\frac{1}{10}x -\frac{14}{5} = 0$$

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

$$r + s = \frac{1}{10} rs = -\frac{14}{5}$$

Two numbers r and s sum up to \frac{1}{10} exactly when the average of the two numbers is \frac{1}{2}*\frac{1}{10} = \frac{1}{20}. 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>

$$r = \frac{1}{20} - u s = \frac{1}{20} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{14}{5}

$$(\frac{1}{20} - u) (\frac{1}{20} + u) = -\frac{14}{5}$$

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

$$\frac{1}{400} - u^2 = -\frac{14}{5}$$

Simplify the expression by subtracting \frac{1}{400} on both sides

$$-u^2 = -\frac{14}{5}-\frac{1}{400} = -\frac{1121}{400}$$

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

$$u^2 = \frac{1121}{400} u = \pm\sqrt{\frac{1121}{400}} = \pm \frac{\sqrt{1121}}{20}$$

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

$$r =\frac{1}{20} - \frac{\sqrt{1121}}{20} = -1.624 s = \frac{1}{20} + \frac{\sqrt{1121}}{20} = 1.724$$

Evaluate

$$10x^{2}-x-28$$