Solve 5x^2-70x+238

$$5 x ^ { 2 } - 70 x + 238$$

Factor

$$5\left(x-\left(-\frac{\sqrt{35}}{5}+7\right)\right)\left(x-\left(\frac{\sqrt{35}}{5}+7\right)\right)$$

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.

$$5x^{2}-70x+238=0$$

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.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 5\times 238}}{2\times 5}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 5\times 238}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-70\right)±\sqrt{4900-20\times 238}}{2\times 5}$$

Multiply -20 times 238.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4760}}{2\times 5}$$

Add 4900 to -4760.

$$x=\frac{-\left(-70\right)±\sqrt{140}}{2\times 5}$$

Take the square root of 140.

$$x=\frac{-\left(-70\right)±2\sqrt{35}}{2\times 5}$$

The opposite of -70 is 70.

$$x=\frac{70±2\sqrt{35}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{70±2\sqrt{35}}{10}$$

Now solve the equation x=\frac{70±2\sqrt{35}}{10} when ± is plus. Add 70 to 2\sqrt{35}.

$$x=\frac{2\sqrt{35}+70}{10}$$

Divide 70+2\sqrt{35} by 10.

$$x=\frac{\sqrt{35}}{5}+7$$

Now solve the equation x=\frac{70±2\sqrt{35}}{10} when ± is minus. Subtract 2\sqrt{35} from 70.

$$x=\frac{70-2\sqrt{35}}{10}$$

Divide 70-2\sqrt{35} by 10.

$$x=-\frac{\sqrt{35}}{5}+7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 7+\frac{\sqrt{35}}{5} for x_{1} and 7-\frac{\sqrt{35}}{5} for x_{2}.

$$5x^{2}-70x+238=5\left(x-\left(\frac{\sqrt{35}}{5}+7\right)\right)\left(x-\left(-\frac{\sqrt{35}}{5}+7\right)\right)$$

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 5

$$x ^ 2 -14x +\frac{238}{5} = 0$$

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

$$r + s = 14 rs = \frac{238}{5}$$

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

$$r = 7 - u s = 7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{238}{5}

$$(7 - u) (7 + u) = \frac{238}{5}$$

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

$$49 - u^2 = \frac{238}{5}$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = \frac{238}{5}-49 = -\frac{7}{5}$$

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

$$u^2 = \frac{7}{5} u = \pm\sqrt{\frac{7}{5}} = \pm \frac{\sqrt{7}}{\sqrt{5}}$$

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

$$r =7 - \frac{\sqrt{7}}{\sqrt{5}} = 5.817 s = 7 + \frac{\sqrt{7}}{\sqrt{5}} = 8.183$$

Evaluate

$$5x^{2}-70x+238$$