Solve 25t^2+24t+4

$$25 t ^ { 2 } + 24 t + 4$$

Factor

$$25\left(t-\frac{-2\sqrt{11}-12}{25}\right)\left(t-\frac{2\sqrt{11}-12}{25}\right)$$

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.

$$25t^{2}+24t+4=0$$

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.

$$t=\frac{-24±\sqrt{24^{2}-4\times 25\times 4}}{2\times 25}$$

Square 24.

$$t=\frac{-24±\sqrt{576-4\times 25\times 4}}{2\times 25}$$

Multiply -4 times 25.

$$t=\frac{-24±\sqrt{576-100\times 4}}{2\times 25}$$

Multiply -100 times 4.

$$t=\frac{-24±\sqrt{576-400}}{2\times 25}$$

Add 576 to -400.

$$t=\frac{-24±\sqrt{176}}{2\times 25}$$

Take the square root of 176.

$$t=\frac{-24±4\sqrt{11}}{2\times 25}$$

Multiply 2 times 25.

$$t=\frac{-24±4\sqrt{11}}{50}$$

Now solve the equation t=\frac{-24±4\sqrt{11}}{50} when ± is plus. Add -24 to 4\sqrt{11}.

$$t=\frac{4\sqrt{11}-24}{50}$$

Divide -24+4\sqrt{11} by 50.

$$t=\frac{2\sqrt{11}-12}{25}$$

Now solve the equation t=\frac{-24±4\sqrt{11}}{50} when ± is minus. Subtract 4\sqrt{11} from -24.

$$t=\frac{-4\sqrt{11}-24}{50}$$

Divide -24-4\sqrt{11} by 50.

$$t=\frac{-2\sqrt{11}-12}{25}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-12+2\sqrt{11}}{25} for x_{1} and \frac{-12-2\sqrt{11}}{25} for x_{2}.

$$25t^{2}+24t+4=25\left(t-\frac{2\sqrt{11}-12}{25}\right)\left(t-\frac{-2\sqrt{11}-12}{25}\right)$$

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 25

$$x ^ 2 +\frac{24}{25}x +\frac{4}{25} = 0$$

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

$$r + s = -\frac{24}{25} rs = \frac{4}{25}$$

Two numbers r and s sum up to -\frac{24}{25} exactly when the average of the two numbers is \frac{1}{2}*-\frac{24}{25} = -\frac{12}{25}. 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>

$$r = -\frac{12}{25} - u s = -\frac{12}{25} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{25}

$$(-\frac{12}{25} - u) (-\frac{12}{25} + u) = \frac{4}{25}$$

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

$$\frac{144}{625} - u^2 = \frac{4}{25}$$

Simplify the expression by subtracting \frac{144}{625} on both sides

$$-u^2 = \frac{4}{25}-\frac{144}{625} = -\frac{44}{625}$$

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

$$u^2 = \frac{44}{625} u = \pm\sqrt{\frac{44}{625}} = \pm \frac{\sqrt{44}}{25}$$

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

$$r =-\frac{12}{25} - \frac{\sqrt{44}}{25} = -0.745 s = -\frac{12}{25} + \frac{\sqrt{44}}{25} = -0.215$$

Evaluate

$$25t^{2}+24t+4$$