Solve 5t^2-21t+20=

$$5 t ^ { 2 } - 21 t + 20 =$$

Factor

$$5\left(t-\frac{21-\sqrt{41}}{10}\right)\left(t-\frac{\sqrt{41}+21}{10}\right)$$

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.

$$5t^{2}-21t+20=0$$

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.

$$t=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 5\times 20}}{2\times 5}$$

Square -21.

$$t=\frac{-\left(-21\right)±\sqrt{441-4\times 5\times 20}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-\left(-21\right)±\sqrt{441-20\times 20}}{2\times 5}$$

Multiply -20 times 20.

$$t=\frac{-\left(-21\right)±\sqrt{441-400}}{2\times 5}$$

Add 441 to -400.

$$t=\frac{-\left(-21\right)±\sqrt{41}}{2\times 5}$$

The opposite of -21 is 21.

$$t=\frac{21±\sqrt{41}}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{21±\sqrt{41}}{10}$$

Now solve the equation t=\frac{21±\sqrt{41}}{10} when ± is plus. Add 21 to \sqrt{41}.

$$t=\frac{\sqrt{41}+21}{10}$$

Now solve the equation t=\frac{21±\sqrt{41}}{10} when ± is minus. Subtract \sqrt{41} from 21.

$$t=\frac{21-\sqrt{41}}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{21+\sqrt{41}}{10} for x_{1} and \frac{21-\sqrt{41}}{10} for x_{2}.

$$5t^{2}-21t+20=5\left(t-\frac{\sqrt{41}+21}{10}\right)\left(t-\frac{21-\sqrt{41}}{10}\right)$$

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

$$x ^ 2 -\frac{21}{5}x +4 = 0$$

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

$$r + s = \frac{21}{5} rs = 4$$

Two numbers r and s sum up to \frac{21}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{21}{5} = \frac{21}{10}. 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>

$$r = \frac{21}{10} - u s = \frac{21}{10} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 4

$$(\frac{21}{10} - u) (\frac{21}{10} + u) = 4$$

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

$$\frac{441}{100} - u^2 = 4$$

Simplify the expression by subtracting \frac{441}{100} on both sides

$$-u^2 = 4-\frac{441}{100} = -\frac{41}{100}$$

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

$$u^2 = \frac{41}{100} u = \pm\sqrt{\frac{41}{100}} = \pm \frac{\sqrt{41}}{10}$$

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

$$r =\frac{21}{10} - \frac{\sqrt{41}}{10} = 1.460 s = \frac{21}{10} + \frac{\sqrt{41}}{10} = 2.740$$

Evaluate

$$5t^{2}-21t+20$$